Yes, as Marcos said.
It would be:
"The car needs
to be white" and
"The symbol needs
to be different from the contamination symbol".
As the conjugation in that verb tense in English goes like that...:
I need (eu)
You need (tu)
We need (nós)
Well, grammatically speaking, that is there´s nothing wrong with the sentence "The car needs to be white".
But, to convey the idea in natural English and a matter of style (my style here) I would redo the wording. I would say:
The car has to be white.
The car should be white.
I need that car painted white.
The car paint work needs to be white.
That way, to me, may lead the reader to think about a car "necessity", it needs. Anyway, you can use the word for the time being, if you want to stick to the grammar point (of using the "s" or leaving it out).
Thank you Artful for your tip.