What is meant by "thy comfort long, and lose thy love thereby"?
We cannot understand the meaning of the lines that you are asking about unless we understand the theme of the whole sonnet. We must also look at the previous line to understand the context in which the phrase “thy comfort long” occurs.
In Sonnet 14, Elizabeth Barrett Browning (or the narrator of the poem) is saying how/why she wants to be loved. She says that she does not want to be loved for merely earthly reasons like “her smile” or “her way of speaking gently.” She fears that those things can change and that her lover will stop loving her if they do. The line you ask about is the last line of the part of the sonnet where she is saying how/she does not want to be loved.
In the previous lines, (9-11) she says that she does not want him to love her because he can wipe her tears away when she cries. She explains that she would not like that because
A creature might forget to weep, who bore/ Thy comfort long, and lose thy love thereby!
What this means is that she is afraid that she might not cry if they were together for a long time. She would be happy because he had loved her for a long time (this is where “thy comfort long” comes in). Because she was happy she would stop crying (“forget to weep”) and that would make him stop loving her (she would “lose thy love thereby”).
So, it is easier to understand the line you are asking about (Line 12) if you look at it in combination with Line 11. Then, you can see that “thy comfort long” refers to him loving her a long time and “lose thy love thereby” refers to what will happen if she “forget(s) to weep.”