Can foreign nationals give blood in Serbia?
Yes, but only after you have lived in Serbia, registered, for at least six months. Besides that, the same health criteria apply as for everyone else, and the doctor at the donation decides.
The six months
The Red Cross of Serbia says it as a condition: a foreign national can give blood if their registered stay in Serbia is at least six months old and they meet all the other criteria. ITKS and ZTKV say the same from the side of the desk: a foreign national who has been resident in Serbia for more than six months shows, besides an identity document, a document of temporary residence. So a visitor, a tourist or somebody who has just moved to Serbia cannot give yet.
What to bring
An identity document with your photo, and with it your document of temporary residence in Serbia. Citizens of Serbia show an identity document with a photo and their personal identification number, and the rest is the same for everyone: what to bring.
What the law says
The Law on Transfusion Medicine does not mention foreign nationals. It asks the donor for written consent before each donation and the questionnaire for the donor's personal identification number, among other data (articles 27 and 29). The six months come from how the services work, not from a provision of the law, so if you are in doubt about your own papers, ask the service you are going to before the trip.
The health criteria are the same
Age 18 to 65, at least 50 kilograms, and a doctor's check on the day: see who cannot give blood. ITKS publishes the donor questionnaire in English too, so you can read the questions beforehand, and the app's Can I donate? screen asks the same ones.