The Dam should last hundreds of thousands of years, even without electricity working.
Interestingly enough the HS channel show life after people addresses the life of the dam and it wouldn't last as long as you might think.
If the dam isn't producing electricity it's designed to pass water through side spill ways. Eventually those spill ways in less than 100 years would get clogged up with mussels that reside in Lake Meade behind the dam. After 10,000 years of water going through the spillways, the dam's concrete structure would collapse.