Swallowing bits of food that are left in the mouth during the day

Undoubtedly
eating is one of the things that invalidate the fast. Allaah says
(interpretation of the meaning): 

“and eat
and drink until the white thread (light) of dawn appears to you distinct
from the black thread (darkness of night)”

[al-Baqarah 2:187] 

It is well
known among the Muslims that fasting means refraining from eating, drinking
and intercourse, and all other things that invalidate the fast. 

Majmoo’
Fataawa Shaykh al-Islam, 25/219 

Eating means
deliberately introducing any substance into the stomach via the mouth. 

See:
Haashiyat Ibn Qaasim ‘Ala al-Rawd al-Murbi’,
3/389 

It is not
conditional that this eating be something beneficial or that the amount be
large, rather if a person swallows something that does not benefit him (such
as a pearl) or he swallows a small amount of something, then he has broken
his fast. 

Swallowing
bits of food that may be left between the teeth is regarded as eating, so it
invalidates the fast. 

This applies
if the fasting person swallows it by choice, and is able to expel it but he
swallows it deliberately. But if it has already reached his throat and he
swallows it and cannot expel it, then there is no sin on him and his fast
remains valid, because in all cases where a thing invalidates the fast, that
is conditional upon the fasting person doing it by choice. If he does it by
force or involuntarily, then his fast is valid and there is no sin on him. 

We have
explained that in the answer to question no.
22981. 

Ibn Qudaamah
(may Allaah have mercy on him) said in al-Mughni, 3/260: 

If a person
has food between his teeth, one of the following two scenarios must apply: 

1 – It is a
small amount that he cannot spit out, so he swallows it. This does not
invalidate his fast, because it cannot be avoided. It is like saliva. Ibn
al-Mundhir said: The scholars are unanimously agreed on that. 

2 – It is a
large amount and he can spit it out. If he spits it out there is no sin on
him, but if he swallows it deliberately, his fast is invalidated according
to the majority of scholars, because he has swallowed food that he could
have spat out willingly when he is mindful of his fast. So this breaks the
fast just as if he deliberately started eating. End quote. 

To sum up
the answer: 

If he is
able to expel it but he does not do so and he swallows it, then he has
invalidated his fast. If he swallows it involuntarily, then his fast is valid and there is no sin on him. 

And Allaah knows best.