Looking into this a bit further, the internet informs me that synthetic formic acid is a fairly common industrial chemical, used most prominently as an additive in cattle feed and as the major active ingredient in household lime remover (e.g. CLR). I think it's much more likely that the rope involved in the mountaineering accident had suffered from exposure to some kind of product like this, rather than natural formic acid secreted by insects.
Surely there is nothing to realistically fear from ants (or wasps) on your rope, unless perhaps they build a big nest in it, in storage-- in which case there would probably be more physical than chemical damage anyway.