This Vox article by Dylan Matthews cites these two studies, which try to get at this question:

• Bendavid & Bhattacharya (2008) — finds 1.2m deaths averted in twelve countries from ~2004–7
• Bendavid et al. (2012) "found that PEPFAR reduced all-cause adult mortality in affected African countries by 16 to 20 percent."

EDIT to add: here's a more recent analysis, looking at mortality impact up to 2018 — Kates et al. (2021)

Since writing the question, I found this study estimating the impact of PEPFAR during its first decade. It concludes that the program resulted in 11,560,114 life-years gained (p. 3). Rough linear extrapolation from the chart on p. 5 (though note that growth was superlinear in the reference period) would suggest an additional 25 million or so life-years were gained between 2014 and 2021, vindicating the "tens of millions of life-years" Open Phil estimate.

Dylan Matthews points to another study finding 1.2 million deaths averted by PEPFAR by 2007. Matthews points out that naive extrapolation from this finding would suggest 6 million deaths saved by the end of 2018 (when the Vox article was written), also noting that the true figure is probably higher because of the study's focus on a limited number of partner countries and because of significant growth in PEPFAR's funding over time. In drawing this inference, Matthews links to a webpage on the PEPFAR's website. The link now redirects to PEPFAR's homepage, but a Wayback Machine search reveals the contents of the original URL: a press release announcing that "Latest Results Show PEPFAR Has Saved Over 17 Million Lives". However, again no source is cited for the estimate. The press release does mention a report, but the archived report I found had no explanation for its estimates of lives saved.