So far, not really. You need some basic AWS experience for security/scaling/setting up alerts purposes, but beyond that the only "ops" I've done in two years has been manually triggering some upgrades (paranoia on my part) and we ran out of disk space at one point as I forgot to set up an alert but this was very easily fixed.
No one on our team has DevOps or DBA experience -- we feel good about Heroku PG (even as inflated prices).