Generally, yes. Are you a contractor (schedule C) or W-2 employee?
The rules for travel are covered in IRS publication 463, chapter 1.
https://www.irs.gov/forms-pubs/about-publication-463
As an employee, your employer can reimburse you tax-free for work-related travel that is temporary, meaning it is expected to last, and actually does last, less than 1 year. Travel that lasts more than 1 year, or that has an indeterminate end date, is not reimbursable tax-free. Also, as soon as you decide to extend your assignment past 1 year, the reimbursement is immediately taxable from that point–it doesn't only become taxable after 366 days, it becomes taxable as soon as the expected end date is more than 1 year or is indeterminate.
(In other words, an assignment that is indeterminate, but actually lasts less than a year, was never deductible/tax free in the first place, because it was indeterminate. An assignment that is expected to last less than 1 year, that actually lasts more than 1 year, becomes non-deductible/taxable on day 366 or on the day the expectation changed, whichever comes first.)
The employer can continue to reimburse you, but they must include the cost of your housing, travel, food and other benefits as part of your W-2 taxable wages.
If you are self-employed, you must report all your compensation, even if it is for housing. You can then deduct the housing costs on schedule C as a travel expense. Again, as soon as the expected end date is more than 1 year, or as soon as the expected end date becomes indeterminate, you lose the ability to deduct food and lodging as a travel expense.
