If you are departing with visual conditions - you can leave it auto tuned.
If you are departing IFR on a standard SID, you need to monitor the aids that SID is based on below the MSA (in the same way that you are required to tune approach aids for an instrument approach when below MSA, even if flying as an overlay, with LNAV selected).
If you are departing IFR on a P-RNAV (RNP 1) SID - and trained/approved etc you can leave it auto tuned.
I personally don't bother tuning aids for an engine failure, as it is such a remote possibility - unless it was IFR in the hills somewhere and I thought I could need the aid before I could ask for it to be tuned.