20 Amp Power Cables into a 15A Power Conditioner

None at all assuming the plugs fit. :) 

You can always go thicker than spec, but not thinner. :) 

You might need plug adaptors if the 20 amp cables use 20 amp plugs. Amazon has got them.

Thanks for the advice.

Depending on the quality of your components and the dollar amount you want to spend you might want to contact VooDoo Cables for a cryogenically treated adapter. I have found these to be very good without altering the sound. Break in is most likely required.
https://www.audiogon.com/listings/lis9334e-voodoo-cable-15-amp-to-20-amp-iec-adapter-cryo-treated-ac...
VooDoo also makes a 20 amp to 15 amp IEC adapter.  

If you use an adaptor it will be detrimental to your sound just use the right connector!!

I used Voodoo adaptor with zero sonic issues.

If you use an adaptor it will be detrimental to your sound just use the right connector!!

That's my intention.  I was offered a good deal on 20 amp cables and was just looking at options.