Does medicine which is swallowed (either dry or with water), but not at all chewed need a kosher hechser? Is there a responsa on this?
First of all, Consult Your Own Local Rabbi (CYOLR).
Star-K writes with regards to taking non-kosher tablets:
In tablets, the most common possibly non-kosher ingredients are magnesium stearate, calcium stearate, and stearic acid. These inactive ingredients serve as lubricants that allow the medicinal powders to adequately “flow” through the processing equipment. The stearates also make it easier for the finished tablet to “release” from the machinery at the final compression stage where the tablet is formed. These stearate ingredients may be derived from either animal or vegetable sources. However, under normal circumstances, enough lubrication will be provided with less than 1.6% stearates, making them batel b’shishim and permissible. Therefore, even one who has a maychush or who is in good health may take tablets if the only kashrus concern is whether or not the stearates are batel.11
If the tablet is definitely non-kosher or contains ingredients that may be non-kosher (e.g. polysorbate 80, triacetin), a sick person – even though he is not in danger – may take it shelo k’derech achila.12 However, one who has a maychush and finds it necessary to take such a non-kosher medication should consult a Rav.13 The same is true for a sick adult or child who is not in danger but requires non-kosher flavored chewable tablets (i.e. k’derech achila).
According to this opinion, if you have a fever, and the only pill that has non-kosher ingredients, it is allowed to take it. Refer to here.
With regard to the situation of "Maychush and Baree" - slight discomfort - it is encouraged to take kosher-medicines. If this is not possible, consult your Rav.
Note that taking pills that are eatable (flavour), it is not allowed if it is non-kosher, since this is considered eating (refer to situation 3).
For preventive reasons, it is sometimes allowed, only if it is taken in a unusual manner. It reads:
If a person is healthy or has a slight discomfort and his physician advises that there is a high risk of him becoming a choleh sheyaish bo sakana or even a choleh she’ain bo sakana, this person may take non-kosher supplements shelo k’derech achila, in an uncommon manner.
Needless to say, preserving life takes precedence over keeping kosher. When there is even some (non-negligible) risk of loss of life, such as in the above cases, a person is permitted to take even tasty non-kosher medicine.