Is it permissible to buy a house by entering a draw and paying non-refundable fees?
Refraining from continuing with this program is the right thing to do. In fact it is obligatory, because entering this draw by paying a non-refundable fee is getting involved in a kind of gambling. The definition of gambling is that into which a person enters by means of risking some money, and he will either lose this money or he will win.
Based on what you mentioned, it is clear that the one who does not win this draw will lose the fees that he paid; if his name is picked then he will be a winner -- with regard to buying a house. Thus it is clear that getting involved in this program is haraam.
This program is similar to getting involved in government or private auctions, for which one pays fees in order to participate, and usually what is taken from those who pay admission fees to the auction is much greater than the value of the booklet or brochure that is given to them. The Islamic Fiqh Council has banned paying more than the price of the booklet or brochure which contains the conditions and guidelines for the auction that is planned. We have quoted the text of the statement in the answer to question number 2150, in which it says: According to sharee’ah, there is nothing wrong with charging an entry fee to cover the costs of producing booklets listing regulations, etc., so long as it is not more than the actual cost.
What we advise you is not to buy a house through this program.
And Allah knows best.