This is actually more common among women than you think. My response is a basic one. If you have a tampon remaining longer than 1 day, you should call your doctor, and get clearence from your doctor. This goes for if you removed it yourself, or need a doctor to remove it...the whole purpose of contacting your doctor is to make sure you have not developed an infection via blood.
you do not have to have symptoms to have an infection, and the tampon does not need to be in for very long to develop an infection, especially if you have underlying conditions that you may not even be aware of, like a yeast infection, which is an overgrowth of normal flora, and when this happens, you may or may not get the itchies/scratchies, and may or may not get a discharge that looks like cottage cheese in the walls of the vagina, you may or may not have an odor(not a disgusting one, but one that's not normal for you). Yeast infections have a wide range, from a mild form (where only a doctor would notice, if you are having an pelvic exam), or moderate to severe, and or chronic types, which are more noticable to the woman, and may need to be treated with antibiotics.
More importantly you could develop a bacterial infection, which is completely different from having a yeast infection. Bacterial infections are just that, an overgrowth of bacteria, but instead of yeast its caused from other things, or you could have vaginitis, which is inflamation of the vagina. Most of the time women walk around with these conditions and not even know they have them, because they don't know what to look for, and or don't always have the classic signs, and like most women, are misinformed when it comes to their bodies, or are too embarrassed to discuss issues like this.
Another thing.....the vagina does have a natural odor, it is not odorless, like some want to believe, it bleeds monthly, it secretes normal and abnormal discharge, it secretes natural lubrication, it secretes female ejaculate, and receives the penis....it has an odor, believe me,....it's supposed to, its not an odorless organ...we don't have any odorless organs in or on our bodies.
Anyway, the only sure way you can make sure your ok is to check in with your doctor. Your doctor can address other concerns you may have, especially if you think the tampon remained for a day, and you find it was there for 3 to 4 days, or wks.
Think old blood, and how it is broken down.....old blood has an odor, period., so you have old blood, and a foreign object inside the vagina for days, or wks at a time, and you think you should take it out yourself?
Even if you do, you should still get cleared, and at least speak with your provider so he/she knows what has happened, and will make the decision on whether or not to put you on antibiotics for a few days.
You also need to confirm that the tampon was removed in tact, and no bits and pieces were remaining.
This is just some basic advice, and its not a big deal only if you ignore it.
We remove tampons all the time, and our patients happen to be medical students, nursing students, graduate students, and fellows/residents, and their spouses....you'd be surprised of how many times this has happened.