"Please do not borrow my dictionary."
The default way it would be written, yet still not accepted, even now, 16 months later... ^^;
That could be the case if the verb were かる. However the verb is かりる（かります）, so the り doesn't end up changing.
The verb is かりる, not かる. Basically it's a godan verb like たべる or みる where you just cut off the る at end for conjugation rather than change it to a ら。It's somewhat confusing and nuanced, and I believe if you're at all interested in the grammar of the language, I'd do research on it
The 私の feels awkward and unnecessary here. Spoken in context, you're just say 辞書.
But it could mean don't borrow the one that we are looking at in the library.
This failed: 私のじしょを借りないでください。
And it said: 私のじしょをかりないでください
It can't read the kanji for borrow or am I wrong ?