According to the Mosaic Law, blood was to be sprinkled on the alters, before the curtains and other places in the temple.
This would result in many places, especially the curtains, being blood stained if not cleansed regularly.
Is there any mention of if and how the temple was cleaned in The Bible, Jewish literature, or elsewhere?
It is logical to think that the priests were careful, so that the blood only landed on the floor, and not on the actual curtains.
Regarding Beit HaMikdash, the whole place was covered with aqueducts and water channels from the surrounding rivers/lakes. These would lead the blood (and other remains) outside.
For example, the Mishna in Yoma 5, 6 talks about the blood flowing away and outside into the Kidron river, and even sold to farmers as fertilizer.
Similarly, the Malbim on Vayikra 16, 14 explains the passuk that says "והזה באצבעו על-פני הכפרת", in which it may seem that sprinkling is actually on the כפרת. Mainly:
מלות "על פני" הוא לפעמים על גבו של דבר ממש... ולפעמים הוא נגד גב הדבר... בא ללמד שלא יזה על צד מערב הכפורת, ממש על גבה, רק על פני היינו נגד הגב. ובזה מטה ההזיה לצד קדם, כנגד הכפורת, והטיפים נופלים בארץ.
There's also a book by Menasheh Harel, with a whole section discussing "המים לטהרה, להיגיינה ולפולחן בבית המקדש בירושלים" (the water for purification, hygiene and worship in Beit HaMikdash in Jerusalem), in which he discusses this as well.
To supplement, not supplant, Cauthon's good answer, I'll note that the mishna (Midos chapter 3) says that the altar and its ramp would be cleaned every Friday with a cloth, because of the blood. (This is Rabi's statement, but the commentaries note that he's explaining and not arguing on the other rabbi in the mishna.)
(It's not completely clear to me whether this was a mere wiping or the application of whitewash, though it seems to have been the former. The mishna there discusses "whitening" with whitewash (lime) twice yearly with a trowel, and "whitening" weekly with a cloth. It's possible, as the Sefaria translator seems to think, that the weekly cleaning was whitewashing also, but I see no indication of such in the commentaries on the mishna and don't assume so; indeed, the Rav says "היו מקנחים אותן במפה", which doesn't sound like they used any substance.)
In addition to the other answers, we have some evidence that blood did land on the veil in the Holy of Holies when the Kohein Gadol (High Priest) sprinkled it there on Yom Kippur, and that they weren't too good about cleaning it.
The gemara in Meilah 17b quotes the sage R' Elazar Bar R' Yose as having seen the curtain in the treasury in Rome, and it had specks of blood on it. This doesn't preclude the possibility of it being cleaned occasionally, but we see that it wasn't wiped away immediately. (The Temple was destroyed 10 months after the last time the Yom Kippur service was preformed.)
Of course, it's natural that this curtain wouldn't be cleaned much, if ever, as no-one else is ever allowed into the Holy of Holies.
