Back in High School, we had 45 minute lunch periods. We generally played Traveller or Twilight 2000... not D&D. But I knew people running AD&D 1E at lunch.
Assuming you have an honest hour for lunch...
If you can play without minis and maps in the edition you use, yes, it's doable. Keep in mind, setting up the maps is 5 minutes by itself.
If you can't play without minis and maps... 4E will be pretty hard. 1 combat encounter per lunch, and maybe a scene or two extra, or if the encounters are small, 2 small combat encounters.
D&D 3E, 3.5E; AD&D 2E w/Player's Option: Combat & Tactics
Again, minis play makes it tough, tho' non-minis play can go pretty quick.
Big battles likely to be two-day fights, most balanced encounters half-a lunch.
AD&D 1E, 2E; D&D OE, BXCMI/Cyclopedia; most retroclones thereof
Minis play again slows things down, but a 45-55 minute session is quite doable. 1-2 combat encounters, plus some RP.
1E or 2E, Again, minis play makes it tough, tho non-minis play can go pretty quick.
Big battles likely to be two-day fights, most balanced encounters half-a lunch or less.
3E: Setup alone is gonna eat some time every day; 5-10min or so. But it plays fast, and is easy to note. If you use overhead pen on page protected cards instead of using counters and card flips, you can speed it up even more. (Hint: using sportscard pages, you put them in facing away from you, so you write on the side with no openings!)
One of the faster playing engines in the industry... same base engine as the old West End Star Wars. Also, PDF available free legally from DTRPG. A 45-55 minute play session can easily handle 3-4 small combats, a big combat, or a good number of non-combat skill based actions.
Tunnels and Trolls
Also fairly fast playing, and it's easy to grab just about any D&D module and run it in T&T. Typical battles run 5-10 minutes if lopsided; 20-30 minutes if actually even odds.
Typical combats run 10-30 minutes for me, using map on whiteboard, loose movement, etc. Often I don't bother with a map at all. 2E said to be a bit faster.
Some Tips on doing it right
Have any maps or handouts drawn out before hand.
Use a magnetic portable whiteboard; use counters, not minis, and use business card magnet mountings to mount the counters. This allows a battle in progress to be left as is for next session easily.
You can make your own magnetic whiteboard with a sheet of steel and white krylon. To grid it out nicely, use white 7/8" tape with 1/8" gaps, run both ways, then spray with gray or black krylon to fill the resultant holes. Don't remove the tape until the spray is dry. Then run a double layer of electrical tape around the edge. If you make 2 or 3, you can have multiple encounters ready on different boards.
Rules issues: handle by email or bbs post.
Level up: handle by email or phone, if practical.
Character sheets: GM hangs onto or has copies. This due to "Absent or Late Player" issues
Starting: start play on time every time. If someone is routinely late, find out who they want running their character until they get there. If someone's absent, hand their character off to someone else. Late is often going to be not their fault.
Damage tracking: have NPC damage cards handy.
Go/Nogo: have people check in at start of day with a text or email. If too few are there to play, text or email half an hour before lunch.
Time: Figure 1st and last 5 min of lunch are lost to travel, and 5m lost to settle-down. Plan accordingly.