Unless mortgage life insurance is purchased, the responsibility of a veteran mortgage passes to the spouse or the veteran's estate in the event of his or her death. There is a continued obligation to make payments, but don't forget the VA's "Leniency Policy" with regard to forbearance for qualified borrowers who fall on temporary hard times.
Mortgage life insurance can take care of this issue once and for all, but it is not offered through the VA. You'll need to find a qualified private insurance company to make these arrangements. The terms of such insurance may vary from agency to agency.
Take the guesswork out of finding a VA Loan provider. Veterans United Home Loans created this site to educate and empower military homebuyers. Regardless of what lender you pick, it's always a good idea to compare and know your options.