A simple reproducible example will show you what is happening. iszero evaluates on the objective expression and returns false, which gets promoted to 0.
julia> using JuMP
julia> model = Model()
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: NO_OPTIMIZER
Solver name: No optimizer attached.
julia> @variable(model, x)
x
julia> iszero(x)
false
julia> @objective(model, Min, iszero(x))
false
julia> objective_function(model)
0
Please ask more targeted questions with simplified reproducible examples.