The gemara says קשין גרים לישראל כספחת. Rashi and Tosafos offer various explanations. Rashi suggests that since they aren't experts in doing mitzvos it will result either in punishment or negatively influencing others. Rashi then cites that some say it is because all jews are responsible for one another - כל ישראל ערבים זה בזה, therefore we suffer for their lack of performance. Rashi rejects this approach because he writes that Geirim were never included in arvus and proves this from a gemara in Sota 37b. Tosafos rejects the proof because it may simply be that we didn't know the number so it couldn't be delineated, but cites the mechilta that does suggest a number.
Rashi in Rosh Hashana 29a explains that the source for one being motzi others in mitzvos that he has already performed is this concept of כל ישראל ערבים זה בזה. Therefore it should follow that geirim who aren't included in arvus cannot be shleluchei tzibur to be motzi others in mitzvos that they themselves are fulfilling. For example, a ger who already hears shofar or made kiddush can't blow or make kiddush for others. The Pri Megadim (eishel avrohom 53:19 at first says that a ger cannot be motzi others based on this rashi, but backs off based on the shulchan aruch that a ger can be a shliach tzibbur.
Perhaps there is a difference between the concept of responsibility for one another, and liability for failing to do for others. The concept of אע"פ שיצא מוציא - being motzi others, is based on the responsibility that all Jews have for one another, even geirim. Since they are part of klal yisroel, anytime another Jew is deficient in a mitzvah, it is as if they themselves are deficient. However, it is possible that Rashi holds that we wouldn't punish geirim for aveiros of others. Responsibility for aveiros committed by others comes from failing to rebuke and influence, which may not apply to geirim since their influence on משפחתות מיוחסות בישראל is limited.