With certain limitations it is permissible. Here is a great link to a brochure on tefillin use from Rav Shimon Eider of Lakewood. He discusses borrowed tefillin on page 43.
It's possible to look at the discussion in Aruch HaShulchan, Orach Chaim, vol. 1, 14:11-14 and conclude that if one is 'unkempt' or 'unclean' that they would be required to ask permission from the owner of a tallit left in a shul. But this is not understanding the language used by the author.
When considering a talit or tefillin left in the shul, it excludes those that have been 'hidden away', meaning where the intention of the owner is communicated through their action of 'hiding' the item. This means that the item is not left in the public domain. Using such a talit or tefillin crosses into the domain of theft.
The other emphasis is that when the item is left in the public domain, the borrower must, as best as possible, return it in the condition that they borrowed it. If found folded then replace it folded, etc. The emphasis is about not causing any loss to the one from whom you borrowed the item.
The Aruch HaShulchan then mentions the subject of those who are "Makpied". He quotes the Tur which says that if you know someone is 'kapied', meaning an owner who is a 'kapdan', this leniency to allow borrowing is not permissible even if their talit is left in the public domain of the shul. It isn't about being clean or free of skin conditions.
To apply that limitation it is necessary to understand what a 'kapdan' is according to its usage in Torah. That is found in Avot 2:5 quoting Rabbi Hillel, "וְלֹא הַבַּיְשָׁן לָמֵד, וְלֹא הַקַּפְּדָן מְלַמֵּד."
Kapdanut is an exceedingly bad character trait which is indicated by an extreme intolerance of others. In general, this type of person is one to be avoided. A discussion of this subject can be found at this link:
The Aruch HaShulchan comments that around him, he observes many people with this character trait and to be careful about it. But again, even with the 'kapdan', it is emphasizing the idea of creating loss to them, real or imaginary. This is the same emphasis he quotes from the Semak and the Mordechai and the Nimukei Yosef. The Aruch HaShulchan enumerates especially 'kapdanim' with a new talit being concerned over perspiration and cleanliness. The emphasis is not on the fact that the borrower perspires. All human beings perspire. The emphasis is on the intolerance of the owner, the kapdanut.
The second term the Aruch HsShulchan uses here is 'nakiyut', which means cleanliness. It is possible, especially from his time period that he could have been including the concept of cleanliness as it relates to health. We have an obligation to protect the body, 'Shemirat HaGuf' from physical harm. Some maladies are transmittable via contact, like for example poison ivy or poison sumac. The oils that cause the irritation can be transferred through contact. So a person with such a condition could transfer the condition to another via the tefillin for example. So too, a person with lice could transfer the lice via a talit or tefillin. A person with such a condition should not borrow a talit or tefillin without the express permission of the owner.
But in the case of eczema, this skin condition is not contagious, even during extreme flare ups. So that concern would not apply. See the attached link.
And to put this personality trait of 'kapdanut' into proper perspective it's worth looking at what appears in tractate Derech Eretz Rabbah, chapter 3.
One who loves his neighbors and is friendly to his relatives, and one who marries off his sister's daughter, and one who loans even a small amount to a poor man in his need, of them the Torah says (Isaiah 58:9) "Then shalt you call, and the L-rd will answer..."