The Weasley twins obtained the Marauders Map some time in their first year at Hogwarts and gave it to Harry in his third year. This means that they canonically had it throughout Harry's first year when Professor Quirrell was the Defense Against the Dark Arts teacher. At the time Quirrell was playing host to Voldemort's soul (or at least a piece of it). Why didn't Voldemort show up on the Marauders Map?
Voldemort's name would have shown up as "Quirinus Quirrell" because he was in Quirrell's body and not his own.
He was sharing a body with Quirrell. He was not in his own body - therefore he wouldn't show up as "Tom Riddle" but instead as "Quirinus Quirrell". And Fred and George wouldn't suspect a thing.
"He is with me wherever I go," said Quirrell quietly. "I met him when I traveled around the world. A foolish young man I was then, full of ridiculous ideas about good and evil. Lord Voldemort showed me how wrong I was. There is no good and evil, there is only power, and those too weak to seek it... Since then, I have served him faithfully, although I have let him down many times. He has had to be very hard on me." Quirrell shivered suddenly. "He does not forgive mistakes easily. When I failed to steal the stone from Gringotts, he was most displeased. He punished me... decided he would have to keep a closer watch on me..."
"See what I have become?" the face said. "Mere shadow and vapor, I have form only when I can share another's body, but there have always been those willing to let me into their hearts and minds. Unicorn blood has strengthened me, these past weeks, you saw faithful Quirrell drinking it for me in the forest, and once I have the Elixir of Life, I will be able to create a body of my own. Now ... why don't you give me that Stone in your pocket?"
Harry Potter and the Philosopher's Stone, Chapter 17
“Then... four years ago... the means for my return seemed assured. A wizard — young, foolish, and gullible — wandered across my path in the forest I had made my home. Oh, he seemed the very chance I had been dreaming of... for he was a teacher at Dumbledore’s school... he was easy to bend to my will... he brought me back to this country, and after a while, I took possession of his body, to supervise him closely as he carried out my or- ders.
Harry Potter and the Goblet of Fire, Chapter 33
He wouldn't show up as "Voldemort" in the first place, but rather as "Tom Riddle".
...and therefore Fred and George would not have recognised him (Harry did not have possession of the Map at this time). We know that, in Pettigrew's and Crouch Jr's case, your real name would still show up on the Map - even if you're in Animagus form or under the effects of Polyjuice potion. Tom Riddle would therefore show up on the Map instead of Voldemort, as Voldemort's merely an alias, whereas Tom Riddle is his actual name.
The majority of the Wizarding World had no idea that the brilliant Head Boy Tom Riddle was actually Voldemort. Even if Tom Riddle did show up "next to" Quirrell on the Map, Fred and George wouldn't have an inkling that something was amiss.
‘Very few people know that Lord Voldemort was once called Tom Riddle. I taught him myself, fifty years ago, at Hogwarts. He disappeared after leaving the school… travelled far and wide… sank so deeply into the Dark Arts, consorted with the very worst of our kind, underwent so many dangerous, magical transformations, that when he resurfaced as Lord Voldemort, he was barely recognisable. Hardly anyone connected Lord Voldemort with the clever, handsome boy who was once Head Boy here.’
Harry Potter and the Chamber of Secrets, Chapter 18
It's likely that JK Rowling hadn't thought of the concept of the Marauders Map when she was writing The Philosopher's Stone, thus leaving out an explicit explanation why Fred and George don't ever see Voldemort on the Map.1
“I solemnly swear that I am up to no good.”
While she didn't directly touch on the subject, JK Rowling did give a similar explanation on the "Pettigrew-Scabbers plot hole".
Fred and George used the map for their own mischief-making, so they concentrated, naturally enough, on those portions of the map where they were planning their next misdeeds. And finally, you must not forget that hundreds of little dots are moving around this map at any given time… Fred and George did not know everyone in school by name, so a single unfamiliar name was unlikely to stand out.
- The Map was so "crowded" with unfamiliar names that another unfamiliar name wouldn't stand out
- They wouldn't have suspected something was amiss based on the two explanations I've given above
There is not a definite answer as JK Rowling didn't touch on the subject (although she did mention why Pettigrew was never seen). However, given the compelling evidence of how the Marauder's Map actually works, this answer "is as canon as it gets".
1 Kudos to TimSparrow who fore-mentioned this here.
At the time, Voldemort was '"Less than a spirit, less than the meanest ghost'. The map was designed to monitor living people who could give The Marauders a punishment, or interfere with their mischief. As the commenters observed, all things have limitations, and this could be a limitation by design.
Also, Fred and George were intent on monitoring Mr. Filch and his cat, also the teachers who could catch them. They could possibly not see a 'Lord Voldemort' or, more likely, 'Tom Riddle' dot, if it was around.
Out-of-universe, I suggest that the map was not conceived at this point.
It's my belief that Voldemort simply wouldn't have shown up on the map. Quirrell is his own person. At that point, Voldemort is just a fragment of a soul latched onto Quirrell.
Luckily, we have a test case to test this theory. Harry Potter is essentially the same as Quirrell: a person with a piece of Voldemort's soul latched on him. We know that Voldemort never shows up on the map next to Harry, so it's safe to say the same would be true of Quirrell while he was possessed.
Other factors to consider:
Voldemort would show up on the map as "Tom Riddle." At best, the twins would simply notice that Quirrell spends a lot of time next to some guy they've never heard of.
The twins pass the map along to Harry because they no longer needed it by their fifth year-- they had memorized the passages out of the castle. They may have only been using the map sparingly by their third year.
Harry had the map for months and noticed that Mad-Eye Moody was Barty Crouch exactly once. He never noticed Peter Pettigrew or Sirius in his third year. If you aren't looking for someone in a castle of thousands, you'd only notice something was odd about them if they came across your path while you were managing mischief.
Seems to me that if Voldemort had shown up at all on the map it would have been as Tom Riddle, a name that would have been meaningless to the Twins. And then the map was used to spy out where people were in relationship to where they were. They as likely would have seen what would have appeared to be Quirrell and Riddle standing, walking or... oh my.... together oh so close. ;) The names could even have been overlapped, and they'd have just thought it a glitch.