David McIlveen, who was a minister with the church for 42 years, said that Free Presbyterians had themselves protested at places of worship in the past.
Mr Poots announced online this week that the family of anti-transgender Christian campaigners had sought him out on Sunday at his place of worship, but that he had already left.
It came a fortnight after the Burke family confronted another DUP politician, Mervyn Storey, at his church, filming the encounter and posting it online.
On that occasion, they accused Mr Storey of endorsing transgender ideology by his chairmanship of the Education Authority (because it has issued guidance advising teachers to essentially accept transgenderism among pupils), and contrasted the authority's stance with Mr Storey's own traditionalist Christian beliefs.
Both Mr Storey and his church, Hebron Free Presbyterian in Ballymoney, rejected the Burkes' accusations and their tactics.
Now Mr Poots too has been targeted.
No footage of the incident has been posted online, and it is not confirmed why the Burkes had chosen Mr Poots.
However, it is likely linked to a recent controversy surrounding Belfast's transgender NHS service for children, which was created while Mr Poots was minister in 2014.
Mr Poots has said he was “never briefed, had sight of, or [gave] approval” for the service.
Speaking in a video posted online, Mr Poots said: "I'd encourage the Burke family if they'd wish to speak to me about political issues to contact our office, make an arrangement, and I'll happily meet them here in my office.
"I'd add that I'd encourage the family to desist from going to church meetings and Gospel services, where the Gospel has just been preached, to doorstep people.
"It's unbecoming, it's unbefitting, and it's not the way things should be done in a church setting.
"If people want to talk politics, they've got all week to do it. I'm here and available to do that.
"So I'd encourage the Burke family to contact me directly and engage with me directly. They don't need to come and stand outside a church building to get me."
Asked about this, Rev McIlveen said: "I fully understand where the Burkes are coming from…
"I also of course can reflect on the fact that me and others were involved in very open and public protests in the past as well.
"Whenever we felt there was an issue that needed to be protested against, we did, and got the applause and acclaim of the church in general.
"And sometimes that led to the disrupting of services and also of course very strong, vocal protests outside churches as well.
"So we've been there - we know exactly what it is to express a point of view in very difficult circumstances."
He added: "I think we have to be wise enough, I believe, and open enough to recognise that people do have a right to express their position, and we've done it before ourselves, and we have to be able to cope with that accordingly."