In R's randomForest, is predict() non-deterministic?
The model will not change unless you re-train it.
The same input sample should always have the same output value for a given model. 
changing the seed, saving and reloading, etc should have no impact on the results.

The training itself is indeed non-deterministic, predict is not. 

library(randomForest)
# sample 80% of data to train
split = sample(1:nrow(iris), floor(nrow(iris) * 0.8))
df_train = iris[split,]
df_test = iris[-split,]
# rf model
mod = randomForest(Species ~ ., df_train)
# predictions
set.seed(123)
res1 = predict(mod, df_test)
set.seed(999)
res2 = predict(mod, df_test)
identical(res1, res2)