What's worse Hyperthermia or pneumonia?

Firstly I believe that you actually intended to ask comparing hypothermia and pneumonia and not hyperthermia which is the opposite.

hypo- is the root for too low for example:

hypothermia = is too low core body temperature

hypoglycemia = is too low blood sugar level

hyper- is the root used for too high for example:

hyperthermia = is too high core body temperature (as in heatstroke)

hyperactivity = is too highly active

If I'm mistaken there I apologize for doubting it but what you described suggests you were thinking of hypothermia not hyperthermia and seems more likely to mix up something as similar sounding as hypo/hyper than cold/hot.

Anyways as for which is worse both can make one seriously ill and both can kill if left untreated so kinda in short they both suck and I wouldn't wish anyone to suffer either to be honest. Beyond that well I guess which of the two would be worse is a matter of personal opinion so best I can really do there is explain and offer some comparison on the two.

Pneumonia

Pneumonia is actually an inflammatory condition of the lungs usually the result of an infection which can be fungal or parasitic though bacteria and viral pneumonia are the most common, inhaled irritants such as smoke, caustic chemicals or particulates can also result in pneumonia. It is not however caused by cold weather this is a common misconception, cases of pneumonia peak in the winter months because of the peak in seasonal infections at that time of year. Mammalian lungs are prime real estate for all manner of bacteria, fungi and parasites and a vital organ to the host as such mammals have evolved a fearsome array of defenses to protect them from infection this makes colonizing the lungs a difficult task as such many of the infectious agents which cause pneumonia are opportunistic hunters taking advantage of the large numbers of weakened hosts provided by other infections most notably such as influenza and especially the common cold with it's rapid waves of devastation through the population while also causing symptoms that rarely incapacitate hosts allowing infections to easily spread as people continue their day to day lives and interact with other people.

Treatment of pneumonia for the most part would be plenty of fluids and taking a few pills to attack the bacteria/virus/fungus/parasite possibly with the addition of some pain medication.

Hypothermia

Mammals including humans are very reliant on the complex systems of thermoregulation which maintain our internal temperature in a fairly narrow range between 36.5-37.5C a reduction in temperature causes a slowing of all chemical reactions which results in lethargy, weakness, poor cognition and generally slows down the biochemical processes that sustain life. For mammals our adaption to deal with the problems of changing temperatures in the environment is our internal thermoregulatory system allowing us to insulate our biological processes from the whims of the external environment. This however means that unlike for example reptiles which have had to evolve to function with a changing internal temperature we lack the enzymes that work effectively outside of that narrow temperature range fairly modest changes in our internal core temperature can therefore quickly become a serious problem. Hypothermia is clinically defined as a core body temperature bellow 35C and there are several stages as the temperature decreases.

Stage 1: Mild Hypothermia (32-35C) - Awake and shivering

Anyone living in a cold climate will be familiar with this one though may perhaps not be aware that the shivering actually is the onset of hypothermia, this is the warning sign the body is still actively compensating thus simple passive rewarming by insulating them with dry clothes and moving to a warm environment their natural ability to generate heat will allow them to recover.

Stage 2: Moderate Hypothermia (28-32C) - Drowsy and not shivering

The onset of moderate hypothermia is marked by the victim ceasing to shiver and becoming increasingly drowsy, the drowsiness is a warning of the inevitable next stage if active rewarming using a heated blanket or in a push a hot water bottle or other direct contact with a warming source of heat to warm them they will begin to rapidly slip into a hypothermic coma.

Stage 3: Severe Hypothermia (20-28C) - Unconscious, not shivering

Should the victims condition proceed to this stage they have slipped into a hypothermic coma, at this point the victim needs treatment in a hospital treatment now would require internal warming by irrigating the body cavities with warm fluids. Treatment of severe hypothermia can result in a rewarming collapse should this occur without medical supervision it could easily prove fatal in itself.

Stage 4: Profound Hypothermia (<20C) - No vital signs

Reaching this stage the victim is now at the stage of being technically dead, hypothermia does slow the process of cellular damage post cardiac arrest and the victim may be able to be revived for some time after their heart stops beating. Treatment of profound hypothermia requires full cardiopulmonary bypass surgery to place the victim on a heart lung machine for extracorporeal rewarming, without a functioning circulatory system to carry oxygen and warmth through the system this is the only method that could warm the body without them spending far too long warm in warm ischemia which would cause widespread brain damage.