In the Catholic Church communion is considered the ultimate sign of unity and thus someone who is not in union with the Catholic Church should not receive communion. However there are exceptions. The Catholic Church does not have a problem with members of the Orthodox Churches receiving communion because their faith is so close to Catholicism (canon 844, §3). Christians who share the Catholic view of the Real Presence may also receive communion for a grave reason if they request the sacrament and express that they share the Catholic beliefs of the Eucharist (canon 844, §4).
You specifically ask if a Presbyterian may receive communion at a Catholic Mass. Presbyterianism does not share the Catholic theology of the Eucharist and therefore would not be considered eligible to receive communion under any of the exceptions in canon 844, §4. So, yes, it would be improper for someone to give them communion. Whether its sinful or not would depend on the attitude of the person distributing communion. They may be doing so out of genuine ignorance (not sinful) or they may be stubbornly ignoring Catholic teaching (sinful).
Nursing home situations are always difficult to determine who is or isn't a Catholic. Not all patients are able to clearly communicate. The general rule of thumb is that if doubt exists then the minister of communion should err on the side of assuming the individual presenting himself or herself for communion is doing so legitimately.
For more information: USCCB's Non-Catholics and Holy Communion