It would be interesting if responders to this questions qualified their answers with a brief explanation that religion plays in their own lives.
Point is, if you’re primarily secular (not practicing, agnostic, atheist, etc) in life and opinion, then I think it’s easy to see where you’d say, “No, it shouldn’t keep you from pursuing that person with a different religion.”
Many religions (notably Christianity and Islam) promulgate the idea that we should not be ‘unequally yoked’ with unbelievers.
For purposes of this discussion, the idea there is that arguably your relationship with God is the most important over all others, and if the two of you cannot agree on that basic precept, then perhaps you’re not really as compatible on other levels as you think.
(disclaimer, not speaking for all Christians, Muslims, etc…. just making an observation from that point of view. Obviously, if your religion of choice forwards the concept of dating/marrying unbelievers specifically, well, then you’d want to take that into account. ;-)