Your can keep your house purchased with an FHA loan and then rent it out and again borrow another FHA mortgage for a new house. But a borrower is only permitted to do this in certain circumstances if he is required to relocate far from a commuting distance for a new job or existing job.
Now the primary residence of the borrower of new FHA loan is the new house. But the situations under which this loan application is approved are limited as per the FHA rules.

You can take out a new FHA loan if the principal residence is evacuated in favor of another principal residence but this is not valid on existing rental properties revealed on the application that has been verified by tax returns as per as the Schedule E of form 1040. The mortgage on the evacuated property has been covered by FHA then you can qualify for a second FHA insured mortgage after undergoing an exemptions described in handbook HUD-4155.1.

Therefore, FHA loans are appropriate for private buyers but will not be a good idea to use for business purpose for purchasing and renting properties.
A homeowner can apply for a loan on a new house and can give the first house on rent only if an individual has a FHA loan with a loan-to-value ratio less than 75%.
You can qualify for a FHA loan after your rental property and income has been evaluated properly. You need to check the other FHA rules to understand what a borrower should do if he rents out a house and buy a new home. Make sure that your net rental income should be equal or more than the projected monthly mortgage payment.

The rules on the multi-unit property rental might vary depending on the situation of the borrower. Therefore, the FHA expect the borrowers to keep their rental units self sufficient.