Probably not. They may be reincarnated.
Mr. Tulip is one half of the New Firm in The Truth (the other is Mr. Pin). He is known for his belief that everything will be fine, so long as he has a potato is his possession. When Mr. Tulip dies, Death interrogates him on his beliefs:
Some of the darkness opened its eyes, and two blue glows looked down
'The --ing bastard stole my potato. Are you --ing Death?'
Jᴜsᴛ Dᴇᴀᴛʜ ᴡɪʟʟ sᴜғғɪᴄᴇ, I ᴛʜɪɴᴋ. Wʜᴏ ᴡᴇʀᴇ ʏᴏᴜ ᴇxᴘᴇᴄᴛɪɴɢ?
'Eh? For what?'
Tᴏ ᴄʟᴀɪᴍ ʏᴏᴜ ᴀs ᴏɴᴇ ᴏғ ᴛʜᴇɪʀs.
'Dunno, really. I never --ing thought...'
Yᴏᴜ ɴᴇᴠᴇʀ sᴘᴇᴄᴜʟᴀᴛᴇᴅ?
'All I know is, you got to have your potato, and then it will be all
right.' Mr Tulip parroted the sentence without thinking, but it was
coming back now in the total recall of the dead, from a vantage point
of two feet off the ground and three years of age. Old men mumbling.
Old women weeping. Shafts of light through holy windows. The sound of
wind under the doors, and every ear straining to hear the soldiers. Us
or theirs didn't matter, when a war had gone on this long . . .
Death gave the shade of Mr Tulip a long, cool stare.
Aɴᴅ ᴛʜᴀᴛ's ɪᴛ?
Yᴏᴜ ᴅᴏɴ'ᴛ ᴛʜɪɴᴋ ᴛʜᴇʀᴇ ᴡᴇʀᴇ ᴀɴʏ ʙɪᴛs ʏᴏᴜ ᴍɪɢʜᴛ ʜᴀᴠᴇ ᴍɪssᴇᴅ?
. . . the sound of wind under the doors, the smell of the oil lamps,
the fresh acid smell of snow, blowing in through the . . . 'And . . .
if I'm sorry for everything . . .' he mumbled. He was lost in a world
of darkness, without a potato to his name. ...candlesticks...they'd
been made of gold, hundreds of years ago...there were only ever
potatoes to eat, grubbed up from under the snow, but the candlesticks
were of gold...and some old woman, she'd said: 'It'll all turn out
right if you've got a potato.
Wᴀs ᴀɴʏ ɢᴏᴅ ᴏғ sᴏᴍᴇ sᴏʀᴛ ᴍᴇɴᴛɪᴏɴᴇᴅ ᴛᴏ ʏᴏᴜ ᴀᴛ ᴀɴʏ ᴘᴏɪɴᴛ?
Dᴀᴍɴ. I ᴡɪsʜ ᴛʜᴇʏ ᴅɪᴅɴ'ᴛ ʟᴇᴀᴠᴇ ᴍᴇ ᴛᴏ ᴅᴇᴀʟ ᴡɪᴛʜ ᴛʜɪs sᴏʀᴛ ᴏғ ᴛʜɪɴɢ. Death
sighed. Yᴏᴜ ʙᴇʟɪᴇᴠᴇ, ʙᴜᴛ ʏᴏᴜ ᴅᴏɴ'ᴛ ʙᴇʟɪᴇᴠᴇ ɪɴ ᴀɴʏᴛʜɪɴɢ.
—Discworld: The Truth
Note that in Mr. Tulip's religious views, such as there are, there does not seem to be anything about an afterlife. According to Death, Tulip "does not believe in anything." The entire extent of his belief system seems to be that possessing a potato will make things "all right," without any clear explanation as to what that might be. As such, I think it would fair to say that Mr. Tulip does not believe in the afterlife.
Death deals with this situation by showing Mr. Tulip the lives of the people he murdered, so that Mr. Tulip does indeed feel sorry for everything. Note that Mr. Tulip also has his potato, after a fashion. As such, Death seems to be doing his best to respect whatever belief Mr. Tulip does have.
Afterwards, Mr. Tulip is offered reincarnation as a chance to atone for his misdeeds.
Death placed the final hourglass back on to the air, where it faded
Tʜᴇʀᴇ, he said, ᴡᴀsɴ'ᴛ ᴛʜᴀᴛ ɪɴᴛᴇʀᴇsᴛɪɴɢ? Wʜᴀᴛ ɴᴇxᴛ, Mʀ. Tᴜʟɪᴘ? Aʀᴇ ʏᴏᴜ ʀᴇᴀᴅʏ ᴛᴏ ɢᴏ?
The figure sat on the cold sand, staring at nothing.
Mʀ. Tᴜʟɪᴘ? Death repeated. The wind flapped his robe, so that it
streamed out a long ribbon of darkness.
'I . . . got to be really sorry . . . ?'
Oʜ ʏᴇs. Iᴛ ɪs sᴜᴄʜ ᴀ sɪᴍᴘʟᴇ ᴡᴏʀᴅ. Bᴜᴛ ʜᴇʀᴇ...ɪᴛ ʜᴀs ᴍᴇᴀɴɪɴɢ. Iᴛ ʜᴀs...sᴜʙsᴛᴀɴᴄᴇ.
'Yeah. I know.' Mr Tulip looked up, his eyes red-rimmed, his face
puffy. 'I reckon . . . to be that sorry, you got to take a --ing good
run at it.'
'So . . . how long have I got?'
Death looked up at the strange stars.
Aʟʟ ᴛʜᴇ ᴛɪᴍᴇ ɪɴ ᴛʜᴇ ᴡᴏʀʟᴅ.
'Yeah . . . well, maybe that'll --ing do it. Maybe there won't be no
more world to go back to by then.'
I ʙᴇʟɪᴇᴠᴇ ɪᴛ ᴅᴏᴇs ɴᴏᴛ ᴡᴏʀᴋ ʟɪᴋᴇ ᴛʜᴀᴛ. I ᴜɴᴅᴇʀsᴛᴀɴᴅ ʀᴇɪɴᴄᴀʀɴᴀᴛɪᴏɴ ᴄᴀɴ ᴛᴀᴋᴇ ᴘʟᴀᴄᴇ ᴀɴʏᴡʜᴇɴ. Aɴᴅ ᴡʜᴏ sᴀʏs ʟɪᴠᴇs ᴀʀᴇ sᴇʀɪᴀʟ?
—Discworld: The Truth
Note that reincarnation is seems to be framed as Death respecting Mr. Tulip's stated philosophy to the best of his ability. Also note that Death is taking a great deal of latitude here—Mr. Tulip did not at all believe in reincarnation, nor that being sorry would affect his final dispensation, nor apparently in an afterlife at all. Death is simply doing the best he can (as illustrated by his put-upon attitude earlier).
Mr. Tulip did not believe in the afterlife, and had not even speculated on what awaited him after he died. It is possible that a non-believer who nonetheless speculated who receive some sort of afterlife, but we cannot say for sure. Regardless, the impression I get from the case of Mr. Tulip is that
- Death can determine someone's fate if they do not express a preference. Death, or perhaps the process of reincarnation itself, seems to determine one's new form. For example, Mr. Pin (who lacked contrition) was reincarnated as a potato, whereas Mr. Tulip was reincarnated as a woodworm.
- Reincarnation may be a sort of default for people who do not have an afterlife.
- Unbelievers certainly do not face oblivion as a default!
As such, if someone died who (like Mr. Tulip) did not believe in the afterlife, and who had no god or other such preference to guide their choice, Death would presumably try to accommodate them as best he could, possibly by offering them reincarnation as an option.
As for someone who wanted a final end, who can say? I'm not sure that would be within Death's power, but I suspect he would grant it if it were.