Addendum: derivations and grammar
Adding onto naruto's answer.
One additional way of looking at this that might help explain why で[は]あります cannot be used with ～い adjectives, is that で (not だ nor です) evolved from earlier にて. This is a hint -- the に is essentially the same as the adverbial に added to な adjectives. Examples:
- 元気だ (where the だ is the copula "is")
- 元気です (where the です serves as both the copula "is", and as a politeness marker)
- 元気である → older form 元気にてある
The corresponding grammatical structures for an ～い adjective:
- 高い (where the ～い adjective all on its own already forms a complete predicate -- we don't need a copula. Saying 高いだ is a bit like saying "it is is tall".)
- 高いです (where the です serves just as a politeness marker)
The く here aligns with the に particle in the older form にてある for ～な adjectives.
To modify ある, both ～い and ～な adjectives must be in the adverbial form, although this is obscured by the contraction にて → で.
(As for why ～な adjectives also require the connecting auxiliary ～て before ある, whereas ～い adjectives do not [*高くてある is incorrect], I haven't found any cogent explanation, so if anyone can add that, I'd be most grateful.)