Background

In 30 Rock S2E14 "Sandwich Day", Tracy's left ring finger is wrapped in white gauze. The bandage is present in every scene in which his hand is visible in this episode, but it is never explained or alluded to in any way. The bandage appears again throughout S2E15 "Cooter", but is gone in S3E1 "Do-Over". Tracy had spent S2E13 "Succession" just writing his porn video game with Frank, and his final scene in that episode shows his finger unbandaged. There is no onscreen reason given for his finger to be wrapped starting in his first scene of "Sandwich Day".

• S2E13 "Succession": No bandage
• S2E14 "Sandwich Day": Bandage
• S2E15 "Cooter": Bandage
• S3E1 "Do-Over": No bandage

Possibilities

I can think of two major possibilities for what is going on here.

1. Real injury

The simplest explanation would be that this was a real-world injury that they just do not want to bring attention to because they had not written anything about it. It does almost seem like Tracy Morgan is often intentionally angling his body/hand so that his finger is not in shot. That would be similar to how James Doohan (Scotty in Star Trek) would hide the fact that he was missing a finger in all but a few rare shots in Star Trek. This could be Morgan trying to hide a personal injury that his character was not necessarily supposed to have.

With that said, it is still very visible in many shots. It may just be that it appears obscured in many shots simply because it is his left hand and Morgan/Jordan seems to favor his right hand, so his left hand is just often naturally not in the shot.

2. Cut cold open

Unlike all other regular episodes, "Sandwich Day" does not have a cold open. The episode just starts off with the musical title sequence that usually plays right after the cold open in most episodes. While researching this episode, I cannot find much of any discussion about why this episode lacks a cold open. The best guess I have seen has to do with the fact that this season was affected by the writers' guild strike that year and how due to that, perhaps there was simply no cold open even written for this episode. On the other hand, perhaps they wrote, filmed, and edited the episode with a cold open, and decided to remove the scene at a point when it was too late in production to replace the scene with something else.

For a somewhat similar situation, consider the Office (US) S6E8 "Koi Pond". This episode had originally aired with a controversial cold open, and was re-edited to use the first post-intro scene as the new cold open instead. All reruns, DVDs, and streaming services just have this edited version of the episode. Perhaps this episode of 30 Rock had a similarly controversial cold open, only it got pulled at the last minute before the episode ever actually aired (unlike that episode of the Office).

Whatever the reason for not having a cold open in the final cut, it is possible that there originally was a scene that had showed Tracy injuring his finger. This would explain why the character's finger was bandaged for a full two episodes.

Conclusion

In the end, my guess is that Morgan had injured his finger in some way and they just did not want to bring attention to the bandage. Morgan probably intentionally tried to angle the bandage out of the shot at times, but he and the director did not find it was very important to put too much effort into making sure he kept it out of shot at all times. It's not all that distracting, and if anybody noticed it, that would be easy for anyone to rationalize that this character could have injured himself. All that said, the lack of a cold open in this episode is still very suspect. The timing is perfect for the potential for a cut scene that had shown Tracy injuring his finger as part of some eccentric shenanigans.

For a somewhat similar situation, consider the Office (US) S6E8 "Koi Pond". This episode had originally aired with a controversial cold open, and was re-edited to use the first post-intro scene as the new cold open instead.

For those curious like myself, the original cold open is here. I enjoyed it but I am also unsurprised that it caused a fuss.

The timing is perfect for the potential for a cut scene that had shown Tracy injuring his finger as part of some eccentric shenanigans.

Probably, I highly doubt they shot each scene/episode in chronological order. If Tracy Morgan did truly have some sort of real world injury then it would be scattered throughout the season. Since it occurred only during two consecutive episodes, that speaks to an intentional choice. Maybe email a writer or something?

Probably, I highly doubt they shot each scene/episode in chronological order.

According to the production codes on each episode on the Wikipedia article for that season, only episodes 11 and 12 were produced out of order.

https://en.wikipedia.org/wiki/30_Rock_season_2

I think for weekly sitcoms like that, that was pretty common.

Even if filmed in order, it's not like they film one a week. They film a season at a time.

So if it was a legit injury, it's very unlikely it required a bandage for exactly one episode.

Way more likely it was cut from the show, or else it would be a smaller injury and just a bandaid, or last multiple episodes.

It was at least two episodes. I didn't scrutinize the entire season for bandages, just the few episodes around the episode where I first noticed it. But it was definitely more than just one episode.

EDIT: Those two episodes were also the last two episodes of the season. It would make sense that it would be healed long before the next season began. For all we know, the bandage would have kept appearing if there had been more episodes in that season.