well soap does corrode aluminum, they are not wrong about that. However, usually it is diluted enough in the water to not make a difference. Now that all these dishsoap companies are selling concentrated formulas it might not be as diluted as it used to be.
At the shop I used to work at we used WD40. I didn't like it but it was cleaner than soapy water and it evaoprated.
What is probably worse for your tires is not the soapy water but the condensation that comes into your tire through the compressed air line and sits inside the rim (it also forms depending on temp changes). This is like soaking your alloy wheels in water all the time.
I used soap not to get the tire on but to get the bead to pop on cause it wouldn't freaking slide up into place when I inflated it. lots of bubbles came out around my valve stem whilst inflating, so I'd say most of it got squeezed out anyway.
I watched a great video on tire mounting featuring a tech from Bridgestone. He used Windex, said it evaporates quickly so you don't have to worry about the tire slipping on the rim, but it's slippery enough to allow the tire to seat onto the bead, which, as Borzwazie has pointed out, is all you need the lubricant for anyway.
Unless it's an ice tire with a liner, in which case, get out the slickest stuff you can find.
Here's a link to youtube for "Blowing Up" tire installation.
There's a whole bunch more on there too. I couldn't find the one I was looking for, I saw it years ago, some guys way up north with HUGE tires "Popping" a tire onto a rim.
Back to the original post topic...
I recently bead blasted my RD350 motor, and then put them through the dishwasher -- with dishwasher soap and everything. Yeah, the one in the kitchen. The combination of hot water and dish soap put a very nice oxidation the the aluminum parts. They went from a bright, freshly blasted look to a more subdued, darker grey patina. This is an oxidized layer on the metal and will help protect it in the future, plus give a long lasting vintage look to it all.
This is different from hand-dishsoap on your rims, but I believe it is safe ... maybe even good for the parts.
Peace and grease