Generally, to deduct mortgage interest the taxpayer must be the borrower and actually pay the interest. However, there is a provision to allow a deduction for an "equitable or beneficial owner." An equitable owner is someone who "owns" the property in substance even if not in name and even if not the borrower on the mortgage. There are several factors involved and the best way to learn about it is to read the tax court cases; here are 2, one where the taxpayer wins and one where they lose.
https://www.ustaxcourt.gov/InOpHistoric/PuentesMemo.Haines.TCM.WPD.pdf
https://www.ustaxcourt.gov/InOpHistoric/phansummary.kerrigan.sum.WPD.pdf
Because marriage creates certain presumptions and grants the spouse certain legal rights (such as, if you got divorced, your new spouse might be entitled to a share of your home even though he is not on the title) I think your situation is an easy call that the equitable ownership rule applies. Your current spouse can deduct mortgage interest that he pays, whether or not you file joint or separate returns.
Note that he will check the box for "I did not receive a 1098" (since any 1098 won't be in his name) and the IRS may issue a notice of deficiency, which you would rebut by arguing for equitable ownership. You will also need proof of the mortgage interest paid, which you might or might not be able to get easily.
And @Critter#2 is correct, this unusual relationship may be a violation of the mortgage terms and the bank may be able to "Call" the mortgage immediately if they find out. You should look into refinancing or discuss the matter with an attorney.
