Probably because you're 1) not delimiting the SQL as a string, and 2) passing in a number and comparing it to a date. You need to delimit it properly.
Code:
CurrentDb.Execute "SELECT * INTO SomeTable FROM Logs WHERE Logs.Date = Date();", dbFailOnError
If you want to pass in a different date for instance:
Code:
CurrentDb.Execute "SELECT * INTO SomeTable FROM Logs WHERE Logs.Date = #" & MyDate & "#;", dbFailOnError
I use CurrentDb.Execute because one advantage of Execute over DoCmd.RunSQL is that you don't get that annoying messages and you don't have to turn on/off the SetWarnings.