Pioneer Elite vsx-14 enough for hdi-3800? Alternatives?

That VSX has to be 20ish years old? So the short answer is yes you'd gain an improvement going with a new surround receiver (i.e. Anthem, Marantz etc). Look for one that has room correction, stable to 4ohm and has good clean power. 

If this is not a part of a surround sound system the I'm with Matt's recommendation of going after a nice integrated amp. Like this one that is barely used (it's mine they're selling for me)

https://tmraudio.com/components/integrated-amplifiers/pathos-logos-mkii-stereo-tube-hybrid-integrated-amplifier-hidac-upgrade/

@szejna 

The two you have listed are a good product and worth your consideration as well as the Yamaha also suggested. 

Take a look at The Music Room if you want to purchase them, they have a return policy so that way you can try them and see which you like.

Ignore my comment on room correction and all three of the suggested amps will give you good power.

@szejna 

The NAD you noted has Dirac built in I believe. Room correction software is a different animal but can make a great difference when use properly. If you aren't techy Dirac can be somewhat difficult to navigate. In that case I'd suggest finding a local dealer and have them do it. 

The MiniDSP is a cool piece but in your set up using a integrated you'd only be able to use their product on the sub channels whereas Dirac can do the full range including your subwoofer(s).

Just something else to think about.