Maybe for some occupations in some cities. No way I could carry all the tools I need with me on a bus that rarely takes me within walking distance to the job site in the cities I live near, if I still lived in one of them.
Sure, for some jobs you will need a car and in that case you bump your prices to match. Since everyone doing this job is hit with the same fee, no one is disadvantaged. While if you work an office job or similar, you can get ahead by not driving and eventually everyone capable of not driving will not drive.
The actual price of fuel does not matter much, only the price you are paying relative to others with the same job title.
It works better for the "poor" people if these taxes/incomes go towards immediate public transit. And work closer to homes (it raises the visible costs of restrictive zoning).
The cost of owning, operating, and insuring a vehicle is a pretty high tax.
Funny thing about public transit too, if you build transit to businesses in walkable communities you don't just help the poor.
Wouldn't the more immediate effect be that the poor stop coming into cities?