Is it normal with cold to have increased heart rate?

Cold: Do you mean cold as in upper respiratory infection? You can have a slightly elevated heart rate when you are ill but it should not be significantly faster. Stay well hydrated and rest will help. If heart rate seems faster than slightly elevated see your healthcare worker.

Heart : Yes, especially more so with fever

HR and fever.: Generally, with a fever your heart rate will increase ten beats per minute for every degree increase of your temperature above your normal temperature. So a fever will raise your heart rate, yes.