Indeed, a value of 0 can be transformed, on the Z-scale, to anything, as 0 is still useful information. If we run a test and calculate Z-scores by global mean and standard dev.:

x
     col1 col2
[1,]    0  435
[2,]    5  346
[3,]    4   65
[4,]    4    3

(x - mean(x)) / sd(x)
           col1       col2
[1,] -0.6073070  1.8444661
[2,] -0.5791257  1.3428389
[3,] -0.5847620 -0.2409501
[4,] -0.5847620 -0.5903982

As kuckunniwid implies, there are other reasons why NaN was produced, likely constant expression values / zero variance. Here, we are going to Z-transform by row in a case where row 1 is all zeros, while row 4 has constant expression of 4:

x
     col1 col2
[1,]    0    0
[2,]    5  346
[3,]    4   65
[4,]    4    4

t(scale(t(x)))
           col1      col2
[1,]        NaN       NaN
[2,] -0.7071068 0.7071068
[3,] -0.7071068 0.7071068
[4,]        NaN       NaN