A security official said some 20 protesters were detained, and a senior police officer hurt in the scuffles, in the Talibiya district of the capital, Cairo.
They were broken up after police fired tear gas.
The clashes come amid mounting sectarian tensions after Muslims set fire to homes owned by the family of a Christian man who reportedly flirted with a Muslim girl.
More than 20 young Muslim men shouting anti-Coptic slogans threw rocks at the demonstrators under a bridge.
The dead man has been identified as Makarios Jad Shukr, 19.
Witnesses said he was shot around 6:30am local time, or 4:30 universal time, as demonstrators tried to approach the proposed site of the new church.
The Egyptian government has decided against a plan to turn a Coptic community centre, which is still under construction, into a church.
Non-Muslims are required to obtain a presidential decree to construct new religious buildings and must satisfy numerous conditions before permission is granted.
Witnesses on Wednesday said that Coptic construction workers were arrested on their way to the site.
Copts account for between six and 10 percent of Egypt's population of 80 million.