If the source is strong enough, you could feel metallic taste in your mouth. A not uncommon occurrence during radiotherapy.
But usually no, you can’t feel radiation directly. If item is radioactive enough, like plutonium, it may feel warm. Ironically, that warms comes from alpha radiation, which is not dangerous unless ingested, so you could hold plutonium with bare hands and be fine, but a radiography source emitting stong gamma you’d not feel anything by touch, while it could do a LOT of damage due to its penetrating properties. After a while, you’d feel a burn where you’re most affected, likely in hand holding it.
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