Heat sugar and water in a 1-quart saucepan over high heat, stirring, until sugar is dissolved, then pour into a heatproof bowl set in a large bowl of ice and cold water. Let stand, stirring occasionally, until syrup is cold, about 3 minutes.
Purée lychees with sugar syrup and lemon juice in a blender until smooth, then force through a fine-mesh sieve into a bowl, pressing hard on solids and then discarding them. Fill cocktail shaker halfway with ice cubes and add lychee purée, vodka, and Cointreau. Shake 15 seconds and strain into Martini glasses.