$$ G(x,y,t)=e^{- x \pmatrix{1&0 \\0&-1} - y \pmatrix{0&1\\1&0} - t\pmatrix{0&-1\\1&0} } $$
where $x,y,t \in \mathbb{R}$.
I would like to Taylor expand G around an infinetisimal change of x,y,t.
In the unitary case:
$$ U(t)=e^{-itH} $$
the result is
$$ U(\delta t) = 1- i\delta t H $$
But, how to I calculate it for multiple variables simultaneously?
What is $G(\delta x, \delta y, \delta t)$?
Call
$$ Z = -x \pmatrix{1 & 0 \\0 & -1} - y \pmatrix{0 & 1 \\1 & 0} - t \pmatrix{0 & -1 \\1 & 0} $$
so your problem becomes
$$ e^Z = 1 + Z + \frac{Z^2}{2} + \cdots $$
if you want to keep up to linear terms (like in your example) then
$$ G\approx 1 + Z = 1 -\delta x \pmatrix{1 & 0 \\0 & -1} - \delta y \pmatrix{0 & 1 \\1 & 0} - \delta t \pmatrix{0 & -1 \\1 & 0} $$