Does the linear probability model require the regressand to be zero/one-valued?
The "LPM" label refers to the structure of the equation, not to the estimator. LPM models can be estimated not only by least-squares methods but also by maximum-likelihood for example.  

As regards the nature  of the dependent variable, we are talking about an affine transformation here.  Let a model be with a binary dependent variable and a single regressor for simplicity,

$$y_i = \beta_0 +\beta_1x_i +e_i$$

with $y_i \in \{0,1\}$. We are then told that the original variable was 

$$z_i = a+(b-a)y_i$$

so $y_i=0\implies z_i =a$ and  $y_i=1 \implies z_i =b$.

Then the model for $z_i$ is

$$z_i = a+(b-a)[\beta_0 +\beta_1x_i +e_i]$$

or

$$z_i = \gamma_0 +\gamma_1x_i + u_i$$

$$\gamma_0 = a+(b-a)\beta_0,\;\;\; \gamma_1 = (b-a)\beta_1,\;\;\; u_i = (b-a)e_i$$

The structure of the model remains the same, an equation linear in parameters, so it is an LPM model still. 

Since $(a,b)$ are presumably known values, after estimating the $z$-model, and obtaining estimates for the gammas, we can recover the estimates for the betas also, if they are of interest.