This is a question I hear a lot and there is a lot of confusion associated with it. The reason being, people are not taught scripture but the customs and tradition of men!
There are actually two different things that most people think of when you present this question.
First, lets examine the dining with unbelievers! Jesus himself dinned with unbelievers.
13 And he went forth again by the sea side; and all the multitude resorted unto him, and he taught them.
14 And as he passed by, he saw Levi the son of Alphaeus sitting at the receipt of custom, and said unto him, Follow me. And he arose and followed him.
15 And it came to pass, that, as Jesus sat at meat in his house, many publicans and sinners sat also together with Jesus and his disciples: for there were many, and they followed him.
16 And when the scribes and Pharisees saw him eat with publicans and sinners, they said unto his disciples, How is it that he eateth and drinketh with publicans and sinners?
17 When Jesus heard it, he saith unto them, They that are whole have no need of the physician, but they that are sick: I came not to call the righteous, but sinners to repentance.
In this instance, Jesus saw a Tax Collector sitting at his place of employment and said, "follow me" and he did. This was an invite to know Jesus and Levi (Matthew) accepted the invitation.
Afterward, Jesus was dinning at Matthew's home and the religious leaders were very indignant about it! They accused Jesus of eating with a sinner, and he was guilty!
Jesus reply was that those who were whole did not need help but those who were sick. He was referring to a spiritual sickness here.
Another palce Jesus was rebuked was in
1 Then drew near unto him all the publicans and sinners for to hear him.
2 And the Pharisees and scribes murmured, saying, This man receiveth sinners, and eateth with them.
Here again we see sinners associating with Jesus and eating in his presence. Here Jesus follows with the parable of the Lost Sheep and the Prodigal Son. Personally, I think the title of this Parable should have been, "The Merciful Father".
At any rate, Jesus explains how precious the souls of men are to His Father! To what extent He will go to win or welcome back one of His creation.
In another example Jesus was invited to dinner by a Pharisee.
36 And one of the Pharisees desired him that he would eat with him. And he went into the Pharisee's house, and sat down to meat.
37 And, behold, a woman in the city, which was a sinner, when she knew that Jesus sat at meat in the Pharisee's house, brought an alabaster box of ointment,
38 And stood at his feet behind him weeping, and began to wash his feet with tears, and did wipe them with the hairs of her head, and kissed his feet, and anointed them with the ointment.
39 Now when the Pharisee which had bidden him saw it, he spake within himself, saying, This man, if he were a prophet, would have known who and what manner of woman this is that toucheth him: for she is a sinner.
40 And Jesus answering said unto him, Simon, I have somewhat to say unto thee. And he saith, Master, say on.
41 There was a certain creditor which had two debtors: the one owed five hundred pence, and the other fifty.
42 And when they had nothing to pay, he frankly forgave them both. Tell me therefore, which of them will love him most?
43 Simon answered and said, I suppose that he, to whom he forgave most. And he said unto him, Thou hast rightly judged.
44 And he turned to the woman, and said unto Simon, Seest thou this woman? I entered into thine house, thou gavest me no water for my feet: but she hath washed my feet with tears, and wiped them with the hairs of her head.
45 Thou gavest me no kiss: but this woman since the time I came in hath not ceased to kiss my feet.
46 My head with oil thou didst not anoint: but this woman hath anointed my feet with ointment.
47 Wherefore I say unto thee, Her sins, which are many, are forgiven; for she loved much: but to whom little is forgiven, the same loveth little.
48 And he said unto her, Thy sins are forgiven.
49 And they that sat at meat with him began to say within themselves, Who is this that forgiveth sins also?
50 And he said to the woman, Thy faith hath saved thee; go in peace.
In this instance we see the Pharisee inviting Jesus instead of Jesus asking the Pharisee to follow him. Jesus accepted his invitation and went to his home to dine with him.
Of course any bible student will know the outcome and why Jesus was really invited to the home of this Pharisee.
Jesus was a very popular person among the people and this Pharisee wanted his presence in his home to bring more recognition to himself!
The Pharisee wanted recognition and he invited all his friends to the dinner. He did not see Jesus as the Messiah but as a teacher. He called him Master. The interpretation of this word "Master" here is "teacher" or "instructor" in the Greek dictionary.
Jesus did set and eat with each of these examples above. I personally think this is an example of how we should treat others who are not part of Gods Kingdom. We should never shun them in a public setting or in their homes and should always be aware of the testimony we can share wth them when we are in these circumstances.
There are so many other places where Jesus ate among sinners as where he turned the fishes and bread into enough food to feed thousands of people. There are other instances where people followed him and his disciples and he fed them. Jesus even said in one place that they didn't follow him because of his words but because of the food he gave them.
Lets address the second thought that comes to most folks mind when this question is ask.
We need to understand that an unbeliever or unrepentant person does not worship God in the sense that a follower of Jesus does!
God is a Spirit: and they that worship him must worship him in spirit and in truth.
An unrepentant person does not have the spirit of God in them and therefore cannot worship God in spirit nor understand the truth of who God is!
I know some folks use Cornelius as an example of an unbeliever worshipping God so, lets examine the story of Cornelius.
Cornelius was a Gentile who did not know God but understood there was a God!
1 There was a certain man in Caesarea called Cornelius, a centurion of the band called the Italian band,
2 A devout man, and one that feared God with all his house, which gave much alms to the people, and prayed to God alway.
Cornelius understood there was a God but he did not know how to worship him. He sought God and prayed to Him. This is why the angel came to him and instructed him to send someone to Joppa and and get Peter so he could instruct Cornelius in the way of salvation.
After Cornelius and all his household was saved and baptized in the Holy Ghost, Peter proclaimed,
34 ....Of a truth I perceive that God is no respecter of persons:
35 But in every nation he that feareth him, and worketh righteousness, is accepted with him.
So we see, those who know there is a God and seek Him will find Him! God will send his witnesses to those who desire them. Cornelius sought God but did not understand how to worship him until after Peter came and preached the gospel of Jesus to him and his household.
Notice, the angel didn't tell Cornelius to go to Joppa and set in on a meeting of the saints. This would have been completely foreign to him and he would not have understood the worship of the saints.
So, can a born again Christian worship God with unbelievers? No, the unbelieving cannot worship God! Jesus didn't worship the Father with unbelievers. He taught them, fed them, healed them, rebuked them, comforted them and done many other miracles and signs among them but he did not worship the Father with them!
So many people do not understand this! The original assembly was not set up for unbelievers to come into it and worship with the Christians! The writer of Hebrews was addressing the born again Jewish believers. These new believes had come under persecution and the writer encouraged them to "Not forsaking the assembling of ourselves together, as the manner of some is; but exhorting one another: and so much the more, as ye see the day approaching." (Hebrews 10:25)
Nowhere in the New Testament does the scripture teach Christians to worship with sinners. Like Jesus, we need to be a friend and even dine with sinners but please understand, the assembly of Heaven does not worship with unbelievers!
This is one of the many reasons the modern church in the USA has become more of a place of entertainment than a place of worship. There are usually more sinners present than true followers of Jesus!
Light has not fellowship with darkness! Righteousness has no fellowship with unrighteousness. The Children of the Kingdom has no fellowship with the children of the world!
Jesus worshipped alone or with his followers only. Many times we find in the scripture where Jesus went out into the hills alone to pray or worship. He never invited any unbelievers to join him.
Just so we don't misunderstand what I am saying, lets define "fellowship". (be partaker of) The root word for this means (to form a union)
So, just like a husband and wife come together to form a union and become one, so is the assembly of Jesus to worship together as one! This eliminates worshipping with unbelievers altogether.
When Paul wrote to the Assembly at Corinth he admonished them to come together as one.
10 Now I beseech you, brethren, by the name of our Lord Jesus Christ, that ye all speak the same thing, and that there be no divisions among you; but that ye be perfectly joined together in the same mind and in the same judgment.
There is a difference between worshipping the Father and dining with unbelievers. By all means use this opportunity of dining with them to tell them about Jesus and the Father.