What value does one set, in general, for the null hypothesis of $\beta_0$ ("Y-intercept")? Say there is a linear regression model to estimate Y, that is:

$Y_i = B_0 + B_1X_i + u$

When testing the Betas of our sample's regression model for significance the null hypothesis for $B_1$ would naturally be set as zero (assuming X has no impact on Y).

However, what would the null hypothesis for $B_0$ be set as?
If it is set as zero, is that not assuming that a certain value is taken by Y in the absence of X, i.e. zero?
In general is there any non-arbitrary setting for $B_0$ we would test against?

Or does the value of $B_0$ vary from case to case?

I am aware that a sampling test is conducted on $B_0$ and as such am curious as to what the exact purpose of this test would be? Is there any actual null hypothesis $B_0$ is being tested against or is the data presented only to illustrate the confidence intervals and other statistics, such as variance?
If you simply want to estimate the values of Y corresponding to particular values of X, you can calculate the regression to obtain estimates of $\beta_0$ and $\beta_1$, and need not be concerned with hypothesis testing.

If you want to test one or more hypotheses, then the choice of hypotheses should be driven by the purpose of your research. A common aim is to assess the relevance of an independent variable by testing the hypothesis that its parameter equals zero, but that is just one choice.  In this case you might want to test the hypothesis that $\beta_1=0$, and you can do so without testing any hypothesis about $\beta_0$.

Where matters possibly become confusing is that regression software often, by default, generates t-values for all the independent variables and the constant. If you want to test the hypotheses that $\beta_0=0$ or $\beta_1=0$, these t-values are useful.  But you can also ignore them if you are not interested in testing those hypotheses.  The estimated parameters don't depend on these hypotheses or t-values, so you can still use them in estimating values of Y.