We can only hypothesize because it's never made explicit why the water of the Mirror should not be touched. The first reference to not touching definitely shines no light on it:
`But come, you shall look and see what you may. Do not touch the water! '
Personally, I've assumed that it could pose a danger to the watcher if the water was touched during an illusion. The reason I think that is in the quote from the Fellowship of the Ring:
The Ring that hung upon its chain about his
neck grew heavy, heavier than a great stone, and his head was dragged downwards. The Mirror seemed
to be growing hot and curls of steam were rising from the water. He was slipping forward.
`Do not touch the water!' said the Lady Galadriel softly. The vision faded, and Frodo found
that he was looking at the cool stars twinkling in the silver basin. He stepped back shaking all
over and looked at the Lady.
In that case, the water is visibly steaming and potentially hot. Frodo may have been scalded by touching it, and if that was the case with this vision who knows what effects other visions may have had.
Aside from that, the other options that make sense would be some kind of temporary or permanent damage to the Mirror, i.e. a touch could break the enchantment permanently or would break the enchantment/illusion for that character, so once that person had touched the Mirror they would no longer see whatever was being shown in the Mirror (maybe less of a concern, since Galadriel still warned Frodo not to touch immediately prior to his scene ending in the Mirror).