Are you planning on running heims or TRE's on this setup?
TRE's (at least the 1-ton chevy ones) use 7/8-18 thread, which is even finer than NF thread.
Here's a reference chart that I use a lot:
http://www.korit.com/tapndrill.htm
According to that, for a 7/8-14 hole, you want 13/16", or .8125 for this. I think your original post was mis-understood in that you're actually wondering about the effects of tapping an OVERSIZED hole vs. an undersized one.
Problem there is your minor diameter on the threads is going to be slightly larger than the thread was designed for. This can mess with the thread fitment, giving you slightly more play in the threaded interface, which can cause premature wear and failure. You said the ID of your tube is .818", which means you're only .006" over anyway, which means the minor diameter is only going to be off by .003" on each side...I'd probably be fine with that, as far as tensile strength goes. Just make sure to have a decent amount of thread engagement, and I'd also run a jam nut to tighten the threads and reduce the slop.
To cut that, you'll probably want to chuck the DOM up in a lathe, and use a tap with a larger chamfer on it - a starting tap would work well. As long as you get the tap started true to center it should thread pretty easily.
For TREs, using the 18-pitch 7/8 thread, the desired hole diameter is .828", or 53/64". This is slightly bigger than your tube, so I'd recommend using a boring bar on a lathe to achieve the proper ID. Only go in as far as you absolutely need to in order to keep the ID as true as possible, and if you have access to a ream of that size, that'd be even better. Use a starting tap for the first few threads, then change to a bottoming tap so that the change in ID won't mess up the leading edge of the tap, and you'll be set.