Generally, footwear is a part of the basic compulsory equipment, i.e. no player can play barefoot:
The basic compulsory equipment of a player comprises the following separate items:
(FIFA Laws of the Game 2015/2016, Law 4 – The Players’ Equipment)
However, that does not mean the player would have to stop playing immediately should he lose a shoe; a player can score a goal even after losing a shoe a few moments before:
If a player loses his footwear accidentally and immediately plays the ball and/or scores a goal, there is no infringement and the goal is awarded because he lost his footwear by accident.
(Interpretation of the Laws of the Game and Guidelines for Referees 2015/2016, Law 4 – The Players’ Equipment)
But he cannot continue playing so, and definitely cannot “assume the risk of injury”. Note that there is an offense of playing “in a dangerous manner” which includes possible injury to the player himself, and which is penalized by an indirect free kick. (But that is purely theoretical, I believe; normally, should the player attempt playing barefoot, the referee would instruct him to correct his equipment.)