Not in general; Taken strictly, RAW has possible exceptions
A monster with truesight can, out to a specific range, see in normal and magical darkness, see invisible creatures and objects, automatically detect visual illusions and succeed on saving throws against them,...
Without specific language otherwise, we must interpret "seeing" in normal and magical darkness to mean regular vision, as solid objects are not producing either normal or magical darkness... but the rest of the portion quoted can be interpreted differently.
There is an edge case for seeing invisible creatures and objects that are behind solid objects. A strict reading of the statement that "A monster with truesight can, out to a specific range, see invisible creatures and objects" is that they can see all such creatures and objects within the range. Note the language in the comparable See Invisibility spell:
For the duration, you see invisible creatures and objects as if they were visible, and you can see into the Ethereal Plane. Ethereal creatures and objects appear ghostly and translucent.
In the case of that spell, it specifies that you see them as though they were visible. This indicates that it follows the normal rules of visibility, simply ignoring the "invisible" status. Truesight does not use such language, and does not establish what may block the ability to see invisible creatures, only the range of vision.
An even stronger argument can be made for "automatically detect visual illusions", given that "detecting" doesn't automatically imply a line of sight, the way that "seeing" does...
According to Jeremy Crawford (discussing whether truesight allows one to see a hidden rogue),
Truesight pierces invisibility, illusions, darkness, and false forms. It doesn't pierce total cover or make you hear better. #DnD
He makes clear, here, that Truesight does not pierce total cover. Therefore, the intended interpretation is that Truesight doesn't allow one to see through real, physical objects that one wouldn't otherwise be able to see through, for any of its effects.
* provided as RAW doesn't sufficiently clarify the edge cases