How low does an INT score have to be to prevent a player character from being able to cast spells? Use items or weapons?
Spell texts seem to suggest that the limit for sentience is 3.
The description of Intelligence in the definition of ability scores says that 3 is the usual minimum for PCs (PHB 7):
The abilities are Strength, Dexterity, Constitution, Intelligence, Wisdom, and Charisma, and they typically range from 3 to 18 for most adventurers.
The Animal Friendship only works on beasts that have an intelligence less than 4 (PHB 212):
If the beast’s Intelligence is 4 or higher, the spell fails.
Likewise, creatures with an intelligence less than 4 are immune to Detect Thoughts (PHB 231-2):
If the creature you choose has an Intelligence of 3 or lower or doesn’t speak any language, the creature is unaffected.
And Awaken only works on creatures and objects with either no intelligence score or an INT of 3 or less (PHB 216)
The target must have either no Intelligence score or an Intelligence of 3 or less.
Thus, it seems like an INT of 3 is the boundary between human-like sentience and bestial intelligence.
This does not prevent players from casting spells or using items
However, to address the body of your question, I could not find any rules that state that a low intelligence score necessarily prevents you from using items or weapons. I also couldn't find any rules that stop low-intelligence creatures from casting spells, though the lowest-INT monster I could find that has spellcasting (technically) is a Quaggoth, with an INT of 6.
The Feeblemind spell is the only game effect I know of that references a creature losing it's ability to cast spells or use magic items which can be potentially linked to changes in ability scores:
On a failed save, the creature's Intelligence and Charisma scores become 1. The creature can't cast spells, activate magic items, understand language, or communicate in any intelligible way. The creature can, however, identify its friends, follow them, and even protect them.
It is ambiguous if the loss of abilities is due to the combined fall in both Intelligence and Charisma scores, a fall in just one of the ability scores, or if it is an additional effect of the spell.
Even a character with no int score or with an arbitrarily large negative int score would still be potentially able to wield weapons and cast spells; Intelligence is only one component of a character's mental stats. In general, abilities that reduce a character's ability scores specify what happens when a score drops suficiently low on a case-by-case basis, but right now 'in general' is 2 out of 3 such abilities.
There's not much reason to assume a low int score should prevent these things, though, unless having, say, a low charisma score would also prevent them. In my personal houserules, a score of 0 in any ability is variously extra problematic, but those are houserules.
A low int score conveys serious negative modifiers to Int-based checks. Consider allowing that, and the resulting player disinclination to invoke those checks, to shape the narration regarding the character's intelligence, rather than mandating they not attempt actions in the first place.