Are there any two inequivalent and irreducible $F$-representations of a finite group $G$ (where $F$ is a field of positive characteristic) having the same characters?
I can surely find an example in which the representations are not both irreducible, but I thought it would be nice to find an example in which both are irreducible.
No, characters of a collection of irreducible representations are distinct and linearly independent. This is Corollary 9.22 in Isaacs book "Character Theory of Finite Groups".
If you drop the irreducibility assumption then, in characteristic $p$, the sum of $p$ copies of any representation has the zero character. Brauer characters, which take values in the complex field even when the representation is in finite chracteristic, were introduced (by Brauer) as a means of distinguishing between any pair of non-isomorphic representations.