Word or phrase for "it won't change anything, but we'll protest anyways"

"Fighting a lost cause" is a common term that expresses a continuing struggle against a foregone conclusion.

lost cause

1. a cause that has been defeated or whose defeat is inevitable.

[Dictionary.com]

I think tilt at windmills may fit the context you are describing :

• Engage in conflict with an imagined opponent, pursue a vain goal, as in Trying to reform campaign financing in this legislature is tilting at windmills. This metaphoric expression alludes to the hero of Miguel de Cervantes' Don Quixote

(The American Heritage Idioms Dictionary)

Protests in which no immediate concessions are expected are sometimes described as symbolic. This sense of the word is defined by oxforddictionaries.com as "significant purely in terms of what is being represented or implied".

A futile protest might suggest that even though they hoped to make a difference, the protesters failed to achieve anything at all. But if they'd staged a...

token protest (About 4,130 results in Google Books)

...the implication is they always knew perfectly well this wouldn't affect policy decisions. The only aim is to publicize the fact that they don't agree with those decisions.