I doubt that you'll get performance boost in PHP. It's a good practice to use them since it takes less space when you store (true/false) settings.
In C/C++, you could use them to get a speed boost when shifting left or right..
ie:
x << 1 // equal x=x*2
x << 2 // equal x=x*4
x << 3 // equal x=x*8
x >> 1 // equal x=x/2
since binary shifting is faster than doing a division or multiplication, it was often use for that. The caveat is that it only works with unsigned integer (no float)
using bit shifting for speed improvement is not recommended anymore since compilers are really more aggressive regarding optimizations. If you add code hints the compiler will choose the best method.
Bitwise operators are really practical. A good example is indicated in the first answer