Is repentance obligatory for someone who swore falsely to cover for a Muslim who committed a grave sin, after having promised not to expose him, especially after others began to spread news of the sin?

If your companion repents of his sin, then there is no harm in you swearing that he did not sin, because one who repents of a sin is like one who has no sin. This can be analogized to what is mentioned in the text regarding matters in which lying is permissible: denying a transgression to cover oneself or another, as long as the other person is not open about it. It is better to use tawriyah (double entendre) and ma'areed (equivocation), by swearing while intending that he did not commit the sin on a specific day or in a particular place, so your intention benefits you and you are not a liar.