Ikea actually plans to become "forest positive": https://www.theguardian.com/sustainable-business/ikea-sustai... The article is from 2012, not sure how far they progressed (true sustainability is difficult). I doubt they use any tropical wood though. Most of their wood supplies are already FSC certified.
Arguably they are one of the leading large companies with regard to sustainability. Most of their wood is fast growing pine, and they switched their entire lighting range over to LEDs way back in 2011
http://www.ikea.com/us/en/about_ikea/newsitem/IKEA_pulls_the...
Compared to the BBC article saying that in 2012 they failed to reach their goal of 35% "preferred sources" (undefined), they currently claim to source 50% of their wood from "sustainable sources" (defined as either recycled or FSC-certified)