From Kitzur Shulchan Aruch 10:22:
מֻתָּר לִקַּח תְּפִלִּין שֶׁל חֲבֵרוֹ גַּם שֶׁלֹּא בִידִיעָתוֹ לְהָנִיחָן וּלְבָרֵךְ עֲלֵיהֶן, כְּמוֹ שֶׁכָּתַבְתִּי בְּסִימָן שֶׁלִּפְנֵי זֶה סָעִיף י"א לְעִנְיַן טַלִּית (י"ד ל' ס"ו).
One is allowed to take someone else's Tefillin even without the owner's knowledge and say a blessing on them, as explained in ch. 11.
There, in discussing that one can take someone else's Tallit, based on a principle that one is happy when someone borrows something for the purpose of a mitzvah (if I understand the underlying principle of what he says. I'm not doing a literal translation.)
Let's say while borrowing the tefillin, the strap tears accidentally. Is the person liable to pay for or replace the strap?
The person is considered a "sho'el" - borrower. Under the normal principles of borrowing, he is liable for all damages. However, in this case, perhaps, the strap was so weak already that it may have torn anyway the next time the owner would have put it on. (I.e. - at the time the borrower took the tefillin, there were no noticeable signs of a weak strap or any other noticeable damage, but something accidental occurred. Chances are, if there was some noticeable weak strap, the person might not have borowed the strap to start, as a torn strap would make the tefillin pasul, and the borrower wouldn't care to take the risk, and would rather have borrowed a sturdier pair.)
Also, if it is assumed that the owner allows anyone to use his tefillin automatically, does the owner in a sense relinquish unknown borrowers from accidental damage as well?