Compare the variances of restricted and unrestricted estimators? Problem
Given a linear model $y_i = \beta_1 + \beta_2 x_i +\epsilon_i, \quad i = 1, \dots, n$
I need to compare the variance ordinary least squares estimator of $\beta_2$ without the restrictions and the variance of ordinary least squares estimator of $\beta_2$ under linear restriction $\beta_1= 0$ (i.e. $\Bbb Var(\beta_2^R)$ and $\Bbb Var(\beta_2^U) $).
Is $\beta_2^R$ unbiased and are there any violations of Gauss–Markov theorem?
My ideas
Intuitively, the restricted variance should be less than the unrestricted variance because the restrictions always damage the flexibility of a model hence reducing the variance.
However, I need a mathematical proof for the problem above.
UPD
With help from comments I have obtained that  $$ Var(\beta_2^{UR}) =  \frac{\sum(y_i - \beta_1- \beta_2x_i)^2/(n-2)}{\sum(x_i - \bar{x})^2}$$ and $$ Var(\beta_2^{R}) = \frac{\sum(y_i -  \beta_2x_i)^2/(n-2)}{\sum(x_i )^2}$$
 A: You can find the general formula for the variance of the regression coefficients on many questions on this site (e.g., here.
To facilitate our analysis, let $\sigma^2 = \mathbb{V}(\epsilon_i)$ denote the error variance.  In a model with an intercept term (i.e., allowing free $\beta_1$) you have the variance:
$$\mathbb{V}(\hat{\beta}_2^U) = \frac{\sigma^2}{\sum x_i^2 - n \bar{x}^2}.$$
In a model without an intercept term (i.e., setting $\beta_1=0$) you have the variance:
$$\mathbb{V}(\hat{\beta}_2^R) = \frac{\sigma^2}{\sum x_i^2}.$$
Both of these results can be derived from the general form $\mathbb{V}(\boldsymbol{\hat{\beta}}) = \sigma^2 (\mathbb{x}^\text{T} \mathbb{x})^{-1}$ using the relevant design matrix $\mathbf{x}$ for the models with/without the intercept term (i.e., with/without a column of ones).
As to your latter question of whether $\hat{\beta}_2^R$ is biased, have a look at the theory of omitted variable bias.  If the true intercept of the model is zero then, intuitively, assuming it is zero should not bias the estimator, and should improve our estimation.  Contraily, if the true intercept is not zero then we would expect that assuming it to be zero might cause some problems.  The formula for omitted variable bias should allow you to write the bias of your estimator as a function of the (unknown) true intercept term.

Some final notes on your working: It is worth pointing out that you are using non-standard notation for the intercept and slope terms in the model --- usually we would denote these as $\beta_0$ and $\beta_1$ respectively.  Another thing to note is that the variance equations you have written cannot possibly be correct, firstly because they include the random variable of interest in them, and secondly because they do not include any reference to the variability of the error term in the model.
