Your question is unanswerable without more information.
different states have different rules to determine who is doing business in that state for filing requirement purposes. Also different rules for who is a full-year resident,or part-year resident or even non-resident.
Are you an employee of the company and get a W-2. In that case your company may be required to file a tax return in the "rental state" (the company would probably be required to registering with the state) and certain states may require it to withhold state taxes and you to also file a tax return. - MAYBE, MAYBE NOT as I stated different states have different rules.
On the other hand if you are self-employed, residency is not necessary determined by the fact you own a home in a state and rent in another.
I suggest you repost your question with the following additional information:
name the states and indicate the one you own a home in
how many days do you expect to work in the state in which you rent and how many days do you expect to work in the state you have a home.
the form of the business you work for, - yourself - no llc, single member llc, a corporation you get a W-2, a partnership - you get a W-2 or are you a partner.
