There are different views to this question, Yes and No.
Yes, You can date a non-Christian
Can a Christian and non-Christian fall in love with each other and date, be genuinely happy and spend great things together? Certainly, yes!
There are actually many Christian/Non-Christian relationship which can be seen as phenomenal by any standard. Moreover, it is possible to follow Jesus and bear fruit in your life even if your relationship is not rooted in Christ.
Some cites Biblical proof on 2 Corinthians 6:14 "Do not be yoked together with unbelievers. For what do righteousness and wickedness have in common? Or what fellowship can light have with darkness?"
However, as long as you keep away from sinful behavior then you are okay.
No, You cannot date a non-Christian
Dating someone who is entirely opposite for you is hard. You may not see it yet but sooner, you will feel that both of you are not building a relationship in any solid foundation.
As a Christian, your life is built to trust and follow Jesus and this is not the case for your lover. You will be tired soon in constantly defending and explaining how you feel about God's calling and later you will disengage with Christ because you deeply love your non-Christian lover.
You need to Pray, seriously and intensely.
God will lead you to the right relationship and if dating a non-Christian is His Will then you cannot question it. However, you need to make sure you are listening to His ways and not keep drawing away from Him. Pray seriously and intensely so God will show you how he wanted things to work out for you.
