Fever is part of the body's inflammatory response. This response is initiated when pathogens attack the cells, tissues, and/or organs. Mechanical injury to the tissues of the body such as a cut, scrape, or open wound can also elicit this response.
The inflammatory response is the body's way of trying to kill the pathogen. The elevation of temperature, or fever, is just one of the mechanisms which make it harder for the pathogen to survive.
Because of the elevated body temperature, one is more likely to become dehydrated. Initially this could cause lowered blood pressure due to a loss of blood volume. However, if the blood is losing water it becomes more viscous meaning the heart has to work harder to pump the blood and therefore increasing blood pressure.
When experience illness that causes fever and inflammation, often individuals will have an elevated heart rate to keep up oxygen demands. Increased heart rate will also lead to increased blood pressure.
Probably one of the biggest factors leading to increased blood pressure as it relates to fever is vasoconstriction. Vasoconstriction is when the smooth muscle of the blood vessels contract and therefore narrowing the lumen (consequently causing the heart to work harder and increasing blood pressure). The reason the blood vessels do this is because it reduces loss of heat cutaneously and therefore allows the body to retain heat, increase body temperature, and hopefully kill the pathogen.