My guess is that my homebrew will be done fermenting while I am out of town. Is there any harmful effects if I don't bottle it until a few days after the fermentation process stops?
As long as your beer hasn't been contaminated with bacteria (which won't happen if it's sealed), there is absolutely no harm in bottling it late. In fact, some (specialty) beers are aged after fermentation and before bottling (e.g. in oak barrels). Nothing changes about bottling--you still add the same amount of priming sugar, for example, and it will still need just as long to age in the bottle before drinking.
UPDATE: After seeing the comment regarding contamination, and researching some more, I believe I agree that my off flavor must be due to wild yeast contamination. My downfall must have been a combination of weak yeast pitched directly from packet without starter and plastic PET carboys with liquid fermlocks that probably either went dry or sucked water back in, contaminating the brew.
There's a lot of information, and people discussing multiple factors regarding this, and I don't know enough of the science to speak to that.
In my experience of 3 years and roughly 30 all-grain brews, I have never, until my last 20 gallons, failed to bottle at about 2 weeks, give or take a few days. My last batch was bottled at a little over 4 weeks, and had a flavor my friends and I now refer to as band-aid flavor, and we've come to expect this from the 3 or so brews we have allowed to sit that long in primary fermentation.
This off flavor mellowed with time, but contributed to my two least favorite brews in my time brewing. The total was 10 gallons pale ale, 10 gallons Belgian strong, both recipes I had brewed before.
I can't defend all the reasons, but in my circle of brewers, we avoid going much more than 2 weeks in primary, all things being equal.
Yeast needs sugar as food to convert to alcohol and also while in frementer contributes a little to carbonation. Leaving bottling too late will affect the secondary fermentation and carbonation process once bottled as the yeast has been starved due to eating all of the sugars. So if there is about 20billion yeast cells in a pack of yeast, then you may find that most have died and there are less there to do the work. Thats my understanding anyway.