I doubt anybody would ever notice, provided you didn't mess with the irrigation water. As far as the treasure...it probably sank to irretrievable depths in the mud a long time ago....LOLJim
A little water is typically not the issue, but rather damage to dikes, levees, canals, diversions, etc. Keeping the infrastructure intact is foremost for any irrigation district. Deliberate damage to irrigation infrastructure is considered a serious crime in rural Idaho. Since Mud Lake, as it is today, is primarily an artificial construct for irrigation water distribution, damage to any structure, even earthen, is a serious issue. Anyone digging without district permission is very likely to get a visit by the local deputy sheriff.
As to the “treasure”, when this tale originated, along with a somewhat smaller lake, there were several smaller ponds in a rather marshy area. Some of those became part of the current Mud Lake in a depression era project. Some have been filled. Some still exist as seasonal ponds. Also, the lake is dredged every few decades to help maintain water retention.
I’ve been looking into this, along with other Eastern Idaho tales, for a few decades. So far, after than visiting the site a few times, and visiting with county employees, I see no reason to pursue this tale, regardless of how much information may be verifiable.