For example, in the sentence 飲んだり食べたりした. Does this たり have anything to do with the -(i)tari ending in Classical Japanese that meant the present progressive (and later semantically shifted to past, and then even later truncated to -ta)? As in われ食べたり = 私は食べている?
It seems quite similar, attaching to the 連用形 (albeit with 音便), and with the same form. The meaning is very, very different though. Are they related?
Yes, it does.
In Classical Japanese, たり was an auxiliary verb but in Modern Japanese, it is a particle. The meaning and usage changed somewhat over time in that presently, it is used only in colloquial speech.
Regarding your example phrase われ食べたり, I need to mention the fact that 食べる is a modern verb; therefore, you should not combine it with the Classical auxiliary verb たり. The Classical versions of 食べる are 食ふ・食らふ and 食（た）ぐ.
