Here is an interesting false friend for those trying to learn German. The English word stress has a number of different meanings. According to the Oxford Advanced Learner’s Dictionary, the word is used for, for example, mental pressure (as in stress management), physical pressure (for example, a stress fracture), emphasis (the boss lays great stress on punctuality), and in the context of pronunciation (the stressed syllable is indicated by an apostrophe).
In contrast, the German language has different words for the above uses. Stress is exclusively used when referring to mental pressure and its consequences, see here. The corresponding engineering term in German is Spannung (hence there is no Stresstensor, but a Spannungstensor). Physical pressure that can lead to fractured bones etc. is called Belastung (the translation of stress fracture is Ermüdungsbruch). And finally, stress in the context of pronunciation is translated as Betonung.
An interesting example of a false friend derived from the word stress is the expression Stresstest, referring to a testing protocol for banks to detect the risk of a breakdown in financially unstable times. This false friend even has its own wikipedia entry.