Yes, people do funny things for the sake of rhyme or to fit words to a tune. In the case that you mentioned, we could I suppose say that 'be' can be seen as evidence of the use of the subjunctive mood. In the past, the subjunctive mood might be used with 'if' and 'when' clauses too in the way it can be used with some 'that' clauses today.
Be is here the subjunctive variant of (indicative) is. Poets don't just scatter subjunctive verbs randomly: the subjunctive mood is, generally speaking, used to create a tentative or speculative effect.
In older English, and in poetry, various kinds of subsidiary clauses used and use subjunctive verbs. Be is, for example, common in conditional clauses (if clauses and whether clauses, for example), where the subjunctive often suggests a more speculative condition.
I don't know the work of Westlife's lyricist well, but if we are willing to be charitable, we could interpret when my heart burdened be as meaning on those rare / possible occasions when my heart may be burdened.