I am filing married/separated. My husband lives in our primary residence and pays the mortgage on the home that we co-own. Am I able to claim any part of the interest as a deduction on my taxes?

On a married filing separate return, you can choose if you can agree to split the deductions.  Otherwise it will be allowed for the person who actually pays it.  On the other hand, a community property state may have specific rules.  They are below for your review:

https://ttlc.intuit.com/replies/3301943

Married Filing Separate  does have disadvantages, you will usually pay more tax on a separate return, the standard deduction is half of what a joint return is and if one spouse itemizes deductions the other must do the same because their standard deduction is zero. You cannot take all the credits you may qualify for, for example.  You cannot take the earned income credit or the education credits, and some deductions and credits are reduced at income levels that are half those for a joint return, such as the Child Tax Credit.

https://ttlc.intuit.com/replies/3288477