It might. I'm not an accountant, so this isn't accounting advice. And, in fact, you should check with your accountant. Many people in the scenario you describe are able to deduct not only the taxes and interest but also maintenance expenses and depreciation of their first house. They generally want a positive cash flow, or at least neutral or minimal negative cash flow (more money coming in than going out from the rental).
However, in some situations like that, accountants will point out that the tax benefit is only one element of the equation. Equally or more important might be cash flow or appreciation. So, don't get too focused on the tax benefit aspect and ignore other considerations, such as cash flow and appreciation.
Recognize, too, that being a landlord has its share of hassles. And even if you turn those responsibilities over to a property management firm, you'll still have some headaches. Plus, property management firms cost money.
So, check with an accountant.