What might happen to a heated glass beaker made from material that does not contain boron?

Borosilicate glass has a low coefficient of thermal expansion, which makes it better for situations in which there are wide, rapid temperature changes. For example, if a standard glass container at room temperature is filled with boiling water, the danger of cracking and shattering is quite high due to the inside surface rapidly expanding while the outside is warming up more slowly. A borosilicate glass such as Pyrex doesn't expand nearly as much as plain glass (about one third as much as regular glass). It can thus survive temperature differentials that would destroy plain glass.

It is for that reason that laboratory glassware is typically made of borosilicate glass.

Based on the premise of the question, a heated glass vessel subjected to rapid cooling (e.g., filled with ice water) is susceptible to shattering into a gazillion shards and slivers of glass.