Do I need to fast if I have a cold?

Thank you for your question, Generally speaking yes. Here is what the peninei halacha (zemanim 7;7) wrote: "Almost everyone develops a headache and feels weak on a fast day, and most people find it easier spending the day in bed than continuing to function normally. Sometimes, a person who is fasting even feels worse than a flu sufferer. Nonetheless, such feelings are not considered a sickness, rather the natural effects of fasting, which will pass within a few hours after the fast is over. Therefore, only one who needs to lie down because of an illness is exempt from fasting. One who suffers from the fast itself, however, must continue to fast even if his weakness causes him to prefer to lie down in bed. Only one who becomes so weak from the fast that he leaves the category of a suffering faster and enters that of the infirm may break his fast". All the best!