To choose between linear or generalised mixed effects model, what is the most important thing to consider? Linear mixed effects models are for continuous variables.
Generalised ones are for non continuous, e.g., binomial.
We have a task in which subjects can get each item correct or incorrect. I'd say that is binomial at the level of subject at least.
Other member of the teams says the most important thing to make this decision is the research question, which is "how many items out of 0 they will get correct at each test", and suggested to treat the variable as continuous and use a linear model.
Also, we got many 0s (almost 80% in the last of 3 tests), so maybe we shouldn't even use binomial but zero inflated binomial. This would be important if we decide to use the brms package for R. Meanwhile, we will use lme4.
What do you think?
 A: 
Linear mixed effects models are for continuous variables. Generalised ones are for non continuous, e.g., binomial.

This is not true. See the wiki page for generalized linear models. E.g., the gamma and exponential distribution are generalized linaer models and both are continuous. The difference is that you allow for other distribution than the normal distribution with generalized linear models. 

We have a task in which subjects can get each item correct or incorrect. I'd say that is binomial at the level of subject at least.

Yes that is binomial data. 

Other member of the teams says the most important thing to make this decision is the research question, which is "how many items out of N they will get correct at each test", and suggested to treat the variable as continuous and use a linear model.

I assume that you have fixed number of $n_i$ trials for each subject $i$. In that case it is a fraction that can have value ${0, 1/n_i, 2/n_i, \dots, 1}$. So you should use the binomial distribution as Dimitris Rizopoulos writes. 

Also, we got many 0s (almost 80% in the last of 3 tests), so maybe we shouldn't even use binomial but zero inflated binomial. This would be important if we decide to use the brms package for R. 

As far as I gather, you have some number of subjects, $k$, who each make some number of guesses, $n_1,\dots,n_k$. Then you model $E(y_i/n_i)$ where $y_i$ is the number of correct guesses from subject $i$. Assuming that you have no covaraites then the model with random effects could be 
$$g(E(y_i/n_i)) = \mu + \epsilon_i,\qquad \epsilon_i\sim N(0,\sigma^2)$$
where $g$ is a link function (e.g., logit), $\mu$ is logit of the probability of a subject guessing correct when the random effect is zero, and $\epsilon_i$ is the random effect of subject $i$. Notice that this model easily yield "a lot of zeroes" if the $\mu$ is sufficiently small and you use the logit link function. Hence, a lot zeroes may not be good argument to use a zero inflated binomial in this case.
A: The number of successes out of N trials is a Binomial distribution. Hence, it seems that you should go for a mixed-effects logistic regression.
