Bardic Inspiration gives a bonus die (size depending on level) to an ally to use for an ability check, attack roll, or saving throw. Can this Bardic Inspiration die be used with Magic Missile? Understanding that magic missiles do force damage?
Bardic inspiration can be used for "one ability check, attack roll or saving throw" (PHB page 53). If the inspiration is from a College of Valor Bard then it can also be used for a "weapon damage roll" (PHB page 55).
Magic missile is none of these, so Bardic Inspiration cannot be used.
Text from the PHB (p53) says, emphasis mine:
... the creature can roll the die and add the number rolled to one ability check, attack roll or saving throw it makes.
If the creature attempted to use their Bardic Inspiration die, there would be nothing to roll it with. The PC says they're going to cast Magic Missile and then they roll a bunch of d4s. No part of that process is an "ability check, attack roll or saving throw", there's just nothing to modify.
The College of Valor bard PHB (p55) has an extra ability on their Inspiration, that may seem like it muddies the waters...
A creature that has a Bardic Inspiration die from you can roll that die and add the number rolled to a weapon damage roll it just made...
Here, it is specifically a "weapon damage roll". So this won't work with attack spells, like Eldritch Blast, and it definitely won't work with Magic Missile.