I'd say Shane pretty much hit the nail on the head with his answer. For most verbs, it is completely necessary to pronounce the "r" in order to avoid confusion with the infinitive or imperative. Take the verb "går" ("to go") in the following three examples: Jag går. (I go/am going.) Jag kommer att gå. (I will go.) Gå! (Go!) In the first example, you cannot drop the "r", as it would sound grammatically incorrect, as if you forgot to conjugate the verb. There is, however, one example that comes to mind where the terminal "r" seemingly disappears in everyday speech, and that is in VSO constructions (such as with questions or conditional clauses) where the verb is followed by the pronoun "du". Compare the follow: Går du? (Are you going?) Gå du! (Go, you!) In the first example, I would say something similar to "gå'ru?", were the "r" and "d" merge to something similar to an alveolar flap (similar to the sound "tt" makes in the American pronunciation of "better"). This makes it slightly difficult to hear the difference between the two, though there definitely is one. That being said, there is one verb that comes to mind that pretty much always loses its "r", and that is "är". As the infinitive form is "vara" and the imperative is "var", there is no risk of confusion. Depending on the dialect, it can be realized as either /eː/ or /ɛː/ in everyday speech. Nobody will misunderstand you if you pronounce the "r", but natives rarely do, unless they are speaking very carefully. Hope that helps. :)

I think this might be due to how you are hearing it, rather than it not being pronounced. In normal speech, that R can be pretty soft and not retroflexed. If you are a beginner, that might make it sound like it is not being pronounced, but there is definitely a difference between the verb form with R at the end, and the infinitive form without it. Personally, I would recommend pronouncing the Rs however you can. It might sound a bit harsh and non-Swedish at first (I've been learning for over a year and mine aren't even that great yet), but hopefully as you learn, listen, and practice more, you can begin to pronounce them more like a Swedish person would. I'm not an expert though. Maybe a fluent person will have a better answer for you.

Congratulations! You have discovered something most Swedes will never know! Indeed, in spoken Swedish, the final /r/ is often left out. It depends on two things: First, the "r" has to be unstressed/short which is always the case with present tense verb endings. Second, the initial sound of the following word must be a consonant. -- " ' " (apostrophe) means a letter is not pronounced -- Examples, "r" + consonant: Jag gillar glass. > Ja' gilla' glass. (Always drop "g" in "jag"!) Han bor på soptippen. > Ham bo' på soptippen. (If "n" is followed by "b" it assimilates to "m".) Det är varmt. > De' ä'/e' varmt. (Final "t" is pronounced in some cases. In this sentence, "t" in "varmt" is pronounced because of grammar but you always drop "t" in "det".) If either of these requirements isn't met, "r" is pronounced between words in sentences. Example, "r"+ vowel: Jag letar efter något. > Ja' leta refter nå't. (If the following word begins with a vowel the "r" becomes the initial letter, kind of. Its difficult to write but I hope you understand what I mean.) Stressed/long "r" is typically written as "rr", and thus this never concerns verbs. Example of "rr" vs "r": Granens barr ligger på golvet. > Granens barr ligge' på golve'/golvet. There are exceptions to this rule. "r" + "d" is probably the most important one to know at first. In Swedish, the letter combination /rd/ usually marks just one sound, [ɖ] voiced retroflex stop, and this is the same both between words and within words. Vem bor där? > Vem bo ɖär? (Vem bo/rd/är?) bord > boɖ (bo/rd/) However, "d" is left out of "du" if it comes after "r". Examples: Vad heter du? > Va' heter'u? Vad gör du? > Va' gör'u? In short, present tense verb + consonant = do not pronounce "r". Present tense verb + vowel = pronounce "r". Since you asked about present tense verbs I tried to not get too complicated. The same basic rules apply to "r" within words but there are more exceptions regarding the pronunciation of "r" in words.