Is it considered copyright infringement for remaking a song using the same lyrics and music notes

Copyright law grants to everyone the right to "cover" a song -- that is, to perform the song publicly or to perform and record the song onto a cd or into .mp3 format. This rule applies to all songs (the words and music) that have already been made public (i.e., already been played somewhere). The owner of the copyright in the song cannot stop you.

You have to pay a license fee, though.

If you're only going to perform the song visit ascap.com, bmi.com, and sesac.com. One of those "performing rights agencies" is the agency that's responsible for collecting performance royalties for the song you have in mind. (You need to figure out which one by querying their song databases).

If you're going to record the song then visit the licensing page for the Harry Fox agency at:
See http://harryfox.com/public/licenseeServices.jsp .

If you're starting a business selling covered songs you need to speak with a copyright or music attorney.

Mr. Ballard has given you a very complete answer. The owner of the song can't stop you from "covering" it as long as it's been published. This is called a "compulsory" license because the owner has no choice but to grant you or anyone else who asks a license, for a fee. The link Mr. Ballard to the Harry Faox Agency is useful because Harry Fox has streamlined these license requests

Legend has it that when liberal activist/actor/siner/songwriter Tim Robbins wrote the songs for his satirical film "Bob Roberts," where he played a conservative politician, the reason he didn't release a soundtrack tpo the film was because he was concerned that real conservative politicans would use the songs, and as long as he didn't publish them, no one else could use them.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.