I've developed a web map using ArcGIS API for JavaScrip version 3.8 and its not compatible with IE 7 or 8. I've advised the people who have reported this to update their version of IE but, if it doesn't involve rewriting the website, I'd like to offer something that can work with these versions.
Here's the website: http://il.water.usgs.gov/data/mchenry/map.php
As you can see, in IE 7 or 8 the map doesn't even load which throws javascript errors.
EDIT
I have added some script that changes the markup and css to handle HTML in IE versions lower than 9 found here: https://stackoverflow.com/questions/5449502/html5-and-css3-for-ie7-and-ie8
Since doing this, I've noticed that the map loads on zoom in/out but none of my points load.
Best Answer
As commented by @blah238 both Internet Explorer versions of 7 and 8 should be supported by the ArcGIS API for JavaScript because the list of supported browsers includes:
From the comments it appears that the source of problems must be something other than just the Internet Explorer version being used.
To progress this further you could contact support.esri.com or ask a new question that becomes more focussed than this one by zero-ing in on any learnings that have come from the comments and since that time.