They might...particularly if they have enough evidence to proceed without your testimony. On the other hand, if you're the sole person levying the charges against him, they probably won't.
For instance, if a neighbor saw your bf kick the front door of your house in and leave with your flat screen...there's a good chance the state will issue on that case. If, on the other hand, you told the police that your bf broke into your house (and he didn't) because you were mad at him...you could fess up and the state may not issue.
Bottom line, it's not really about what you (the complaining witness) wants, it's about what the evidence can prove.
You cannot drop (nor file) charges; only the State can. You can tell them that you do not want to pursue the case but they make the call. And, in Harris County they most frequently do not follow the wishes of complainants.
Whether they will follow your wishes depends on some factors including why he was in the house, whether anything was stolen, whether he was there to hurt you or someone else, whether anyone was hurt, his prior criminal record, etc.
If he is a first offender, they could dismiss or they might offer him a deferred adjudication probation.
Although I have answered the question to try to help you, you should consult with a lawyer in your area in person on the matter. In addition, my answer does not establish an attorney-client relationship between us.
The Harris Co DAs office will go forward with the case against your boyfriend with or without you if they have the evidence. If you initially made the complaint to the police, they can use the officer's testimony without you. if someone else complained, they can use that person. They will consider why you don't want to pursue charges, but the decision is theirs alone. No matter what, so long as the charges are pending and he can't pay a bond, he will be held. hire an attorney for him if they don't want to drop the charges.
Although my intent in answering this question is to aid you in the legal process, my answer does not establish an attorney-client relationship in any way. You should seek the advice and counsel of a qualified attorney in your community to evaluate your legal needs and to advise you. No Attorney-Client Relationship is created without the specific intent of both parties.