What must a Catholic deny in order to no longer be considered Catholic?

If you are going to be technical about it, if you deny any doctrine that has been taught infallibly and irreformably, then you cannot be Catholic. That could be anything from the Assumption of Mary to the death of Christ on the cross. Now some doctrines have been made infallible and some have not; some are just authoritative. Someday the church should make a list of what is infallible and what is not, but after 2,000 years she has not done so. Take Humanae Vitae, the 1968 encyclical on birth control. As the church has communicated it, it is authoritative but not stated infallibly, so believe it or not, you could still be a Catholic and be buried as a Catholic even if you did not believe in Humanae Vitae. Some argue it is infallible through the ordinary magisterium, and it may be, but the church has to affirm that the ordinary magisterium has made it infallible; you and I cannot just assume it. So technically you could still be buried as a Catholic, but you are going to answer to God, because it is an authoritative document, and according to Lumen Gentium 25 you have to obey whatever the church tells you is doctrine, whether it is infallible or not. Would you want to stand before God and argue that you wanted to be Catholic but did not believe what He said you have to? Good luck.