If my opponent has an Abomasnow with "Freezing Disaster" Ability in play, can I use Munkidori's "Adrena-Brain" Ability to move damage counters off one of my Pokemon or would that count as "healing"?
Yes, you can do that. Moving damage counters off a Pokemon is not the same as healing damage (similar as to how moving damage counters onto a Pokemon is not the same as doing damage to it).
Source: TPCi Rules Team (2024-06-27)

Relevant cards

Note: All applicable cards may not be listed below.
SV2_11
SV2_11
SV6_95
SV6_95