Yes iron supplement causes teeth discoloration. Make sure the syrup doesn't come in contact with teeth. Use dropper.
Consult pediatrician to know more methods for consuming iron without damaging teeth.
These may also be early childhood caries (tooth decay), rampant caries, nursing bottle caries. Clean teeth and oral cavity well after every feeding. Give to water to the child.
Clean mouth even after night, midnight feedings.
Avoid bottle feeding.
Kindly go through the blog mentioned in the link below.
Next Steps
"How to Protect Milk Teeth (Child Teeth)?" by Dr.Rashmi Bamane - https://www.practo.com/healthfeed/how-to-protect-milk-teeth-child-teeth-39707/post
Health Tips
Visit MDS pedodontist for clinical examination.