Based on Rava's approach 123b, we pasken that a כלי שמלאכתן להיתר can be moved for a permitted function, for the use of it's place, and even to protect the kli itself. But a כלי שמלאכתן לאיסור can only be moved for a permitted function such as breaking nutshells with hammer (as in the mishna 122b), and for the use of it's place, but not to protect the kli itself.
We usually assume that since a כלי שמלאכתן לאיסור can be moved לצורך גופו, meaning for a permitted use, it can be moved without thinking twice. Even if one would be able to accomplish the same thing with a כלי שמלאכתן להיתר, they can move a כלי שמלאכתן לאיסור for a permitted use. However, it is clear from the gemara that this is not correct. The gemara says that even כלים שמלאכתן להיתר cannot be moved for a very minor use such as the sticks that separate the breads on the shulchan since even if they aren't put there for shabbos, the bread will not get moldy. The gemara also says that they cannot be moved if there is a substitute that doesn't require moving any kli such as the case where they were able to skin the korban pesach as it was suspended on their arms.
The M.B. 308:12 writes that one can only move a כלי שמלאכתן לאיסור for a צורך גופו when there is no other kli that has a permitted use that can be moved for this purpose. In the Sha'ar Hatziyun 13 he cites the source for this directly from our gemara (it sounds like it is his chiddush and not found in other achronim). However, in our gemara we find that these limitations apply even to a כלי שמלאכתו להיתר, not just by a כלי שמלאכתו לאיסור. The M.B. seems to recognize this point and says that if we find these limitation by כלי שמלאכתן להיתר, that they can't be moved if the use isn't that necessary, or if there is a legitimate substitute to do the same thing without a kli, we should certainly apply these limitation to כלים שמלאכתן לאיסור. The problem is that by כלים שמלאכתן להיתר we don't find any limitation at all. The Shulchan Aruch 308:4 says that a כלי שמלאכתו להיתר can be moved for any purpose, but not for no need at all. How then can the M.B. cite our gemara as a source to limit כלי שמלאכתן לאיסור when the Shulchan Aruch for whatever reason doesn't apply these limitations to what the gemara is actually speaking about, namely a כלי שמלאכתו להיתר?
This can be explained based on the Ritva. The Ritva points out that our gemara indicates that not all purposes are considered צורך גופו, only very important uses for which there is no better substitute qualifies. He then asks, how can it be that we can move a כלי שמלאכתו להיתר even to protect the item itself, as Rava says on 123b, but cannot move it for a minor use. Wouldn't a minor צורך גופו use still be more permitted than the moving to protect the muktza item? The Ritva answers that this sugya is clearly going with the approach of Abaye 123b who explains the braisa to forbid moving a כלי שמלאכתו להיתר to protect the item itself. According to Abaye it would be forbidden to move it מחמה לצל and would only be permitted to move it for a significant צורך גופו not just a minor צורך גופו. But according to Rava who permits moving a כלי שמלאכתו להיתר even to protect the item, it is certainly permitted to move it for a minor צורך גופו.
But, even if we reject the limitations of the gemara by כלים שמלאכתן להיתר because of paskening like Rava and permitting things to be moved מחמה לצל, we are only compelled to reject it by כלים שמלאכתן להיתר, not by כלים שמלאכתן לאיסור. By a כלי שמלאכתו לאיסור, even we who follow Rava would apply the limitations and only permit the moving of it לצורך גופו when there is no alternative. Therefore, the M.B. is justified in learning from our gemara a limitation about moving a כלי שמלאכתו לאיסור for a minor צורך גופו even though these limitations are rejected by the Shulchan Aruch by a כלי שמלאכתו להיתר.