The proof of Harnack's inequality using De Giorgi method has a great flexibility and it can be exteneded even to doubling metric measure spaces that support Poincaré inequalities.

The elliptic case has been treated in:

J. Kinnunen, N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces. *Manuscripta Math.* 105 (2001), 401–423.
(MathScNet review.)

The parabolic case has been treated in:

J. Kinnunen, N. Marola, M. Miranda, Jr., F. Paronetto,
Harnack's inequality for parabolic De Giorgi classes in metric spaces.
*Adv. Differential Equations* 17 (2012), 801–832.
(MathScNet review.)

There are many other related results. Just search references and citations of these two paper in MathSciNet.