Based on the information you have shared, I assume that you have 1000 samples of signals which are of length 320, and have 10 different noise realizations each.
I suggest you use a nested for loop to go through the third dimension, filtering among different noise levels for each signal. Something like this should work.
for i = 1:1000
for j = 1:10
New_Sig = SigPlusNoise(i,:,j);
SigIn = New_Sig;
cheby = LPF_ChebyI_V1(SigIn,Fsamp,Order,BW,Rp,ImpPlot,PrntPlots);
New_data(i,:,j) = (cheby);
On a side note, I advise you to initialise 'New_data' variable with the correct size before entering the for loop.