You may be interested in this previous answer of mine which details the background behind the passage of the 22nd Amendment, which implemented presidential term limits.

Yes, there was criticism of FDR for running for a third term, just as there had been criticism of previous presidents seeking a third term. The Republican Party made it part of their 1940 platform:

"To insure against the overthrow of our American system of government we favor an amendment to the Constitution providing that no person shall be President of the United States for more than two terms.

This remained in their 1944 platform, and was still part of the party's platform in 1946 when they won back both houses of Congress. Obviously, there wasn't enough opposition to prevent FDR from winning those third and fourth terms, both of which were won in landslides. But there was enough opposition that they were able to get a constitutional amendment passed not long after. It was a partisan issue, but not a hyper-partisan issue, as there was support for term limits across the political aisle. Again, see the answer linked above for more in-depth information on how and why this came about.