..I agree, swap the or for an and....I rewrote the code below and tested it with a positive outcome.
IF EXISTS(SELECT patient.F00069, patient.F00026, patient.F01298, patient.F01299, patient.F01300
FROM patient INNER JOIN
tumor ON patient.UK = tumor.FK1 INNER JOIN
followup ON tumor.FK1 = followup.FK1 INNER JOIN
hospital ON tumor.UK = hospital.FK2
WHERE (hospital.F00026 = '0' OR hospital.F00026 = '1' OR hospital.F00026 = '2')
AND patient.F00069 = '1'
AND followup.F01298 = ' ' AND followup.F01299 = ' ' AND followup.F01300 = ' ')
BEGIN
UPDATE followup SET followup.F01298 = 'C', followup.F01299 = 'C', followup.F01300 = 'C'
END
ELSE
PRINT 'Nothing to Update'