You can use the "Moment" function (given that variance = 2nd moment minus 1st moment squared):

n = 1;
ExpDist = ExponentialDistribution[\[Lambda]];
TruDist = TruncatedDistribution[{a, b}, ExpDist];
OrdDist = OrderDistribution[{TruDist, n}, n];
mean1 = FullSimplify[Mean[OrdDist], Assumptions -> {b > a && a > 0, \[Lambda] > 0}]
var1 = FullSimplify[Moment[OrdDist, 2], Assumptions -> {b > a && a > 0, \[Lambda] > 0}] - mean1^2 // FullSimplify

$$\frac{(a-b) e^{a \lambda }}{e^{b \lambda }-e^{a \lambda }}+a+\frac{1}{\lambda }$$

$$\frac{-e^{\lambda (a+b)} \left(\lambda ^2 (a-b)^2+2\right)+e^{2 a \lambda }+e^{2 b \lambda }}{\lambda ^2 \left(e^{a \lambda }-e^{b \lambda }\right)^2}$$

For $n=2$:

n = 2;
ExpDist = ExponentialDistribution[\[Lambda]];
TruDist = TruncatedDistribution[{a, b}, ExpDist];
OrdDist = OrderDistribution[{TruDist, n}, n];
mean2 = FullSimplify[Mean[OrdDist], Assumptions -> {b > a && a > 0, \[Lambda] > 0}]
var2 = FullSimplify[Moment[OrdDist, 2], Assumptions -> {b > a && a > 0, \[Lambda] > 0}] - mean2^2 // FullSimplify

$$\frac{(2 a \lambda +3) e^{2 b \lambda }-4 (b \lambda +1) e^{\lambda (a+b)}+e^{2 a \lambda } (2 b \lambda +1)}{2 \lambda \left(e^{a \lambda }-e^{b \lambda }\right)^2}$$

$$-\frac{\left((2 a \lambda +3) e^{2 b \lambda }-4 (b \lambda +1) e^{\lambda (a+b)}+e^{2 a \lambda } (2 b \lambda +1)\right)^2-2 \left(e^{a \lambda }-e^{b \lambda }\right)^2 \left((2 a \lambda (a \lambda +3)+7) e^{2 b \lambda }+e^{2 a \lambda } (2 b \lambda (b \lambda +1)+1)-4 (b \lambda (b \lambda +2)+2) e^{\lambda (a+b)}\right)}{4 \lambda ^2 \left(e^{a \lambda }-e^{b \lambda }\right)^4}$$