In The Fellowship of the Ring, Galadriel warns Sam not to touch the water before he looks into the mirror. Then when it is Frodo's turn, as he is slipping forward she also warns him not to touch the water. Is there a reason for this? Is it touched upon in any of Tolkien's other works?
I have only read the Lord of the Rings trilogy and The Silmarillion and I have not managed to find any other reference to Galadriel's mirror to explain it.
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).
With the whereabouts of all palantir still unknown, one was at least potentially under Sauron's watchful eye, either directly or through his minions. Such close contact could subject viewers further to his will, especially dangerous in the case of the Fellowship's members and to the Ring-bearer in particular. The elves knew to avoid this pull, hence their continued use of the currently accounted-for devices and how they still wielded the Rings that allowed them to maintain their way of life by continuing to subdue Sauron's growing magic to their benefit. This assumes Galadriel's mirror of water is either connected to, or is itself, one of the seeing-stones, fashioned as a dish rather than into a ball such as Saruman uses. Also, physical touch could "smudge" or sully the image, leaving dirt and oils on the surface of the water or in the dish; perhaps even the fingerprint of mundane contact could interfere with the magic.
In the Silmarillion, it is said that the first Elves found water to be sacred because the Ainur Ulmo would use water to travel or send messages to them. It could be that the water poured into the basin is enchanted by Ulmo and it must not be touched for fear of distorting the illusion or breaking the enchantment.
