The damage is minimal for all writing instruments.
In order for somebody to bother doing something that could possibly damage the environment, you have to pay a substantial amount of money. As an example, if you pay $1 for a taxi ride, the taxi ride will be very short (nobody would offer you a taxi ride where $2 worth of gasoline will be burnt, and the taxi ride costs $1). Thus, the damage to environment will be minimal.
Do a simple experiment: take a 10 blank pieces of paper. Weigh them using an accurate scale. Write something on the papers, and weight them again. Do you see an increase in weight? Probably not, unless you have a laboratory scale.
Thus, I would focus on the paper, not on the writing instruments. Paper production creates black liquor, and unfortunately the best way to get rid of it is to burn it, releasing carbon dioxide into the atmosphere. About half of the carbon in wood will be burned. So, even if you take all measures possible to ensure the paper will be buried in a landfill instead of being burned, some amount of carbon dioxide will be released in paper production. Also, wood is a limited resource and using it for paper prevents other uses such as mitigating climate change by directly burying wood underground, or burning it as biomass for energy. Thus, by using paper, you are causing more climate change.
Focus on where the problem is! It's not the writing instrument, it's the paper on which you are writing!
We are very fortunate that paper use is reducing due to computers.