Can I replace a general-purpose electrolytic capacitor with a low-ESR capacitor?
Generally, low ESR can be used to replace general purpose capacitors, but there are situations where the low ESR capacitor could cause oscillation due to the use of a finicky regulator. 

For example, the LM1117 is a very common semi-LDO regulator (mostly because it's cheap and you can get them very easily). 

It has requirements as follows (from the datasheet):


  The ESR of the output capacitor should range between
  0.3Ω - 22Ω


A lot of low-ESR 1000uF parts are better than that (by more than 10:1 in some cases).

You could always add a series resistor to degrade the low ESR parts.