Probably a good idea for you both to talk with your healthcare providers to see what they recommend, and if they think STI testing is a good idea. The "burn" could be related to many things (even apart from STIs) but the only way to really know is to get checked out. HPV does not typically lead to burning sensations.http://www.ashastd.org/phpBB/viewtopic.php?t=672
goes to a discussion about HPV within a relationship and it's one you might find helpful. HPV can most often be managed within a relationship, but it's important to know the facts.
Hope this helps,