No, this is not true every time. It depends on the conditions given.
For example-
Suppose we have a Box A with dimensions 10 x 10 x 1 and Box B with dimensions 5 x 5 x 5.
Box B has a surface area = 2(25+25+25) = 150 and a volume of 5*5*5= 125 cubic units
Box A has a larger surface area = 2(100+10+10) = 240 and a smaller volume = 10*10*1 = 100 cubic units
Similarly take an example of sphere.
Lets suppose the radius of the sphere is 2 cm
So, SA is 4πr² = 4*3.14*2*2 = 50.24 cm²
Volume of the sphere is = 4πr³ /3 = 33.50 cm³
Here also the SA is greater.