While a relatively rare phenomenon, there are numerous examples of languages where the selection of a pronoun depends on the TAM of the clause (I say 'clause' as it is not always the case that it is also marked on the verb).
In Pitta-Pitta (Australian family) all core case-marking is split according to whether it marks future or non-future and this also applies to pronouns. (1a) and (1b) show non-future and future examples respectively:
(1a) WaiRa -nha ngathu thari -nya nhana -yi
heart -nfut.acc 1sg.nfut.erg spear -past 3sg.nfut.acc -prox
"I speared the kangaroo in the heart."
(1b) Tyira -maru -ngu nganyu karnta -ø
boomerang -prop -fut.erg 1sg.fut.erg go -fut
"I'll be going with the boomerang."
(Bowern, C. 2004. 'The origins of tense-based case marking in Pitta-Pitta and Wangkajutjuru'. Australian Journal of Linguistics, 24:2)
There are a number of other Australian languages with similar behaviour. Another example is Gurnu:
(2) bulga -nguna -ngadhu bulga -ngaw -adhu
hit -pres -1sg.pres hit -past -1sg.past
"I hit (present)." vs. "I hit (past)."
Blevins (2004) suggests that this is probably due to a combination of factors, including a fixed verb-pronoun word order and the phonological form of verbs causing sandhi changes on following pronouns. (Blevins, J. 2004. 'Evolutionary Phonology'. CUP.)
Iai (Austronesian family, New Caledonia) also has tensed pronouns, but this is reported to be similar to the English pronouns with (what has been called) propositional marking of tense: "I'll, I'd, I'm, I".
Scottish Gaelic has two forms of the 2sg pronoun, /tu/ and /thu/, the former occurring only with future/present habitual verbs. Similar to the situation with Gurnu, these have arisen through phonological change caused by the adjacent verb, Gaelic having VSO order in the clause.
Hausa (Afro-Asiatic family) and Wolof (Niger-Congo family) also have distinct sets of pronouns for indicating different tense/mood. These are sometimes known as 'temporal pronouns'.
Upriver Halkomelem (Salishan family) has tense marking on nouns and pronouns.
On a slightly different tack, it has been shown that two tense-marking auxiliaries in Panare (Cariban family) derive etymologically from demonstrative pronouns.
(Gildea, S. 1993. 'The development of tense markers from demonstrative pronouns in Panare (Cariban)'. Studies in Language, 17:1)