A rapid and anisotropic modification of the Fermi-surface shape can be associated with abrupt changes in crystalline lattice geometry or in the magnetic state of a material. We show that such an electronic topological transition is at the basis of the formation of an unusual pressure-induced tetragonal ferromagnetic phase in Fe_{1.08}Te. Around 2 GPa, the orthorhombic and incommensurate antiferromagnetic ground state of Fe_{1.08}Te is transformed upon increasing pressure into a tetragonal ferromagnetic state via a conventional first-order transition. On the other hand, an isostructural transition takes place from the paramagnetic high-temperature state into the ferromagnetic phase as a rare case of a "type-0" transformation with anisotropic properties. Electronic-structure calculations in combination with electrical resistivity, magnetization, and x-ray diffraction experiments show that the electronic system of Fe_{1.08}Te is instable with respect to profound topological transitions that can drive fundamental changes of the lattice anisotropy and the associated magnetic order.