Is it permissible for an employee residing in government-provided housing, furnished by his employer, to rent out a portion of it to a friend for a fee, given his financial need and outstanding debts?

If you received housing from your employer as part of your entitlements, then it is permissible for you to accommodate anyone you wish with you, whether for a fee or without. This is because you are in the position of a lessee, and a lessee has the right to sublease what they have leased to others. This is the view of the Shafi'is and Hanbalis, and no dissenting opinion is known regarding it.

However, if the housing is a donation from the employer, then it is not permissible for you to accommodate anyone with you except with their permission. If they grant permission, then it is permissible for you to do so, even for a fee, unless it was stipulated that you should not take a fee.