For example, can you say, the toothpaste is little minty, or little fresh? Or, for example, that man is little tall?
You can use little as an adverb to modify an adjective in some cases.
It’s not particularly common, outside of a few stock phrases, to do so with an adjective in the positive, and none of the examples you give sound idiomatic. They’re (probably) grammatical, but I’d say they would normally only be used in a humoristic or sarcastic manner.
The most common collocation I can think of where little modifies an adjective is little known (also sometimes written as a compound, little-known).
Apart from this quite limited usage, there is a more widespread and productive construction where little is used to modify comparatives of adjectives:
Politicians are little more than a bunch of thieves!
Of course, when used this way, little means ‘only a little bit’, so something which is little known is obscure, rare, recondite. It emphasises that it is almost not known at all.
In contrast, what you seem to be looking for in your example phrases is a little, which is (among other things) an adverb that modifies adjectives and means ‘to a limited degree’. If the toothpaste is a little minty, it means that there is a bit of a minty flavour to it, enough to be noticeable. If it is little minty (although that’s an awkward phrasing), you’re implying that you were expecting it to be minty, perhaps because it has pictures of mint leaves and says “Extra minty!” on the packaging, but in reality, there’s almost no mintiness at all.
Tall is an interesting case, because you wouldn’t be very likely to say that someone is a little tall at all—even though it is perfectly logical notionally (nothing unusual about “He’s a little short”, “He’s a little fat”, or “He’s a little skinny”). You would instead say something like, “He’s fairly tall” or “He’s somewhat tall” or “He’s quite tall”.
Using little alone here makes it even odder. “He’s little tall” would mean that he is actually short (whereas “He’s a little tall”, while uncommon, would mean that he is slightly above average height), but it is something you’d only expect to hear by someone who’s deliberately using unusual phrasing for effect, probably humoristic understatement. It’s a bit similar to saying that someone is ‘vertically challenged’.
You would use "a little".
It's a little cold today.
Sorry, I'm a little late.
I'd be careful about using "a little tall", however. It doesn't quite sound right due to the contradictory terms used. Go with "quite tall" in that scenario.