We have found the problem. One of the pages with the Report Viewer had garbage collection disabled. As a result the server was overloaded with the tmp files. After we re-enabled it, we don't see this problem anymore.
protected void Page_Unload(object sender, EventArgs e)
{
if (rpt != null)
{
rpt.Close();
rpt.Dispose();
GC.Collect(); // forces garbage collection
}
}