Yeah .. I can confirm that something is messed up either with the DH one-click installer or with Joomla 1.5.1 on DreamHost.
After "upgrading" a working Joomla! !.5.0 "one-click" install with the one-click upgrader, I get the same results as you, and the others have reported.
"Cleaning the Cache" also did not appear to help - though reading through all the errors produced, it looks like session handling is also involved.
Whether it is the DreamHost one-click installer for 1.5.1, or Joomla! I have not yet determined. I think I'll try to do a manual install of 1.5.1 on my DreamHost server to try to see if that works.
One other thing: my "upgraded" 1.5.0 version still says it is running 1.5.0 in the admin panel, though it is exhibiting the error condition reported to be in version 1.5.1.
This makes me wonder if DH has their distribution files borked between versions or if there is some breakage in the Joomla package for 1.5.1 - it should be reporting itself as 1.5.1, so something is definitely wrong and it will take some more investigation to get it sorted.
I'll post back with what I find after a manual install of 1.5.1 on DH.