Can Likelihood Be 0?

Likelihood must be at least 0, and can be greater than 1. Consider, for example, likelihood for three observations from a uniform on (0,0.1); when non-zero, the density is 10, so the product of the densities would be 1000. Consequently log-likelihood may be negative, but it may also be positive.

Can the likelihood function be zero?

If you have a very inadequate model such that at least one discretely (or continuously) distributed observation has zero probability (or probability density) for any parameter value θ∈Θ, that is, you essentially observe something that is impossible under that model, then yes, your maximum likelihood would be zero.

Can a likelihood be negative?

Probability densities are non-negative, while probabilities also are less or equal to one. It follows that their product cannot be negative. The natural logarithm function is negative for values less than one and positive for values greater than one.