Here's the part we're talking about. In various places it's been called the "barrel cover", the "collar", the "barrel", the "barrel sleeve", the "outer barrel sleeve", the "hood", or just the "black part":

It struck me as an interesting question.
The hole in the cover is only about 6.5 mm diameter while the field width is at least 5.6 mm (at 4X on an APS sensor) and the entrance pupil ignoring the cover is 10 mm (based on NA 0.1 at 4X with a 200 mm tube lens).
Given all these conditions, the hole provides unobstructed access to the center of the field, but it's not big enough to provide unobstructed access to the whole field of view. There will be some vignetting going on. This will certainly darken the corners, but does it have any other good or ill effects?
Based on some experiments, the answer is "not much other effect". The tradeoff is only between better center-to-corner uniformity with the cover removed and slightly less veiling glare with it in place. Removing the hood also increases the working distance from 25 mm to 32.3 mm.
Here is an example of the center-to-corner uniformity issue. This is with the objective pushed down to 2X on an APS-sized sensor, using as tube lens a Canon EF 100mm f/2.8 L IS USM macro lens focused at infinity, with an adapter stack of 67-to-52, 52-to-42, and 42-to-RMS. This is a screen grab from Photoshop with no levels adjustment, just the way the images came out of Zerene Stacker DMap.

I'm not bothering to show detail crops of any areas because I can't see any other visible differences other than brightness (from the vignetting) and veiling glare (from loss of the lens hood effect when the cover is removed).
Most likely I'll be using this objective with its cover removed in the future. There's a slight increase in veiling glare due to stray light getting to the lens, but the veiling glare is much easier to correct with a levels adjustment or a separate hood than is darkening of the corners. Removing the cover also gives about 7 mm more working distance.
--Rik
Edit: to insert picture of the objective & hood.
Edit: to document increased working distance.
Edit: add backlink to earlier discussion.