I should have known you were going to ask.
I will bet on up. Remember when Western Europe moved to the Euro? A disproportionate share of retail prices went up, leading to the designation "the Teuro." ("Teuer" means "expensive" in German.) It seems that retailers had wanted to increase their prices in the first place, but were afraid of irritating their loyal customers. The regearing of the monetary unit gave an "excuse" for price increases plus not everyone noticed the higher prices in the new monetary unit. I predict similar results, albeit smaller ones in absolute terms, from abolishing the penny.
How bad an outcome would this be? Ironically it was Greg Mankiw who wrote of excessively high prices, by a small degree, leading to large welfare costs for the economy as a whole. But this model may not apply to abolition of the penny.
Under one scenario, prices go up but they would have gone up sooner or later anyway. Within a year or two, inflation has caught up with the price increase. In the long run the whole thing is more or less a wash, although we do suffer from higher prices and higher deadweight loss for just a little while.
Under a second scenario, prices go up and remain at a permanently higher plateau. Future price decisions are taken from this new reference point. For this model to work, we must assume that price is a signal of quality and that the frame of reference for interpreting the meaning of a price is based upon an observed status quo. So the price boost comes, everyone assumes that is just how much food (or whatever) is now worth, and that is our new marker for judging future price movements. Keep in mind that these assumptions cannot be true globally (there cannot be Walrasian slack at every margin), but only have to be true across relatively small price increases (N.B.: many tricks lie in here, since the price increases will be large in percentage terms for some goods).
I would bet my money on the first scenario, as I assume Greg Mankiw would as well. If you believe in the second, you probably shouldn’t want to abolish the penny.
You can modify these scenarios in many ways, including through the explicit recognition of option value. Do you know of any empirical tests on which model of prices is the better guess?