I can't find any evidence that plastic straws can trigger an allergy. Plastic straws are very popular, so this might say something. Or it may not.
Depending on the metal, these can trigger a nickel allergy, which is a pretty common allergy (though people vary in sensitivity).
Healthline explicitly mentions "stainless steel cooking equipment and eating utensils" as something that can contain nickel and trigger a reaction on their page for nickel allergy.
Plastic phobia is upfront about it: "Our steel straws contain nickel which is great for corrosion resistance so do take precautions if you are allergic to nickel."
Some people believe some paper straws contain gluten. Even some organizations believe that there are paper straws that contain gluten (and there's a response there from a person who claims to have reacted to the gluten in a certain type of paper straw).
I don't know if any paper straws have gluten in them (and even then, if it would be enough to cause a reaction), but it does look like there are safe paper straws. The National Celiac Association hasn't found any paper straws that contain gluten. For a specific example, Aardvark Straws says that their straws "have not been found to have any known allergens. They are used in various hospitals because of this." (Also, Aardvark makes bendy paper straws, so the graphic isn't entirely accurate in this respect either.)
Acrylates in acrylic nails can cause an allergic reaction, but I don't know about straws as I haven't found any solid proof one way or the other.
I can't find any evidence of bamboo straws specifically causing problems, but I did find evidence that some people have a contact reaction to bamboo. See the sources here: Thermo Scientific. Also: Contact dermatitis caused by bamboo.