The draft law was introduced to parliament on Tuesday and proposes to tax internet providers for each gigabyte of data traffic at a rate of 150 forints, or £0.38.
A statement by the protest organisers said: "The move... follows a wave of alarming anti-democratic measures by Orban that is pushing Hungary even further adrift from Europe."
The group has given the government 48 hours to withdraw the legislation and have vowed further protests if their demands are not met.