I obtained an F1 visa in the United States. In the process, I have received a I-797 (USCIS Approval Notice) for my F1 reinstatement:
Is there any point in keeping the I-797, or can I safely throw it away?
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You should keep it.
For example, one of the requirements for Adjustment of Status (the process of getting US permanent residency from within the US) is that you've never been out of status in the US in any past trips (except if you are an Immediate Relative of a US citizen in which case you don't have to be in status). So if there's even a remote possibility of you applying for this (for example, if you will be in the US on a work visa and apply for permanent residency from the US) at some later point in your life, you want to make sure to retain all I-20s and I-797s to prove you were continuously in status.