When Moshe Rabbeinu is told by Hashem at the burning bush that he should go and inform Bnai Yisrael that Hashem has spoken to him and will soon redeem them from slavery, he reacts by stating that the people will not believe that he's telling the truth (שמות ד:א'). Hashem immediately responds by giving Moshe two signs that he may show the people to prove the veracity of his claim; as part of the second sign, Moshe's hand becomes afflicted with Tzoraas (שם פסוק ו'). The Gemara in Shabbos (דף צ"ז.) understands that this affliction was not merely a random sign for Moshe to use, because, as Rav Achai Gaon explains in the She'iltos (שאילתא מ'), Hashem could have selected any number of other signs. Rather, he chose a sign which contained a lesson, indeed a punishment, for Moshe himself because he had suspected Bnai Yisrael of not believing him. The Gemara thus derives from this story that one who is חושד בכשרים, that is, he unjustly suspects innocent people, is punished with a physical affliction as Moshe was.
The Mishnah in Yoma (דף י"ח:) states that as part of the preparation for the Avodah in the Beis Hamikdash on Yom Kippur, the elders among the Kohanim would have the Kohein Gadol swear that he would not alter the service in any way; following this, both the Kohein Gadol and the elders would cry. The Gemara (שם דף י"ט:) explains that he would cry because they even suspected him of being a Tzeduki (coming from that group of people who do not believe in the validity of the Torah Sheb'al Peh or the authority of the Rabbanan), and they would cry because if they were indeed being suspicious of an innocent man, they would be deserving of the above cited punishment which is visited upon one who is חושד בכשרים. The Rambam (פרק ד' מהל' תשובה הלכה ד') lists חושד בכשרים as one of the Aveiros which prevents a person from being able to fully do Teshuvah, explaining that people do not even realize that it is an Aveirah at all to consider a good person to be a sinner; people will therefore rarely even attempt to do Teshuvah for this Aveirah. It is clear from the above that it is prohibited to suspect an innocent person of being a sinner.
Does this prohibition apply to one's attitude towards all people, or is it possible that sometimes one may indeed be suspicious of someone else? The Rambam, in discussing the case of the Kohein Gadol (פרק א' מהל' עבודת יום הכפורים הלכה ז'), implies that it is prohibited to suspect anyone whose actions and motivations are not known, because perhaps he has nothing wrong in mind. In his Peirush on the above Mishnah in Yoma (פרק א' משנה ה'), the Rambam likewise writes that it is forbidden to suspect someone whose actions are unclear of being bad; the Tosafos Yom Tov on that Mishnah accepts this as well. This position appears to work out very nicely with that of the Mishnah in Pirkei Avos (פרק א' משנה ו') which states והוי דן את כל האדם לכף זכות, teaching that one should judge all people favorably, a trait which the Rambam (פרק ה' מהל' דעות הל' ז') says must be possessed by a Talmid Chacham. Rashi there asserts that unless one knows otherwise for sure, one should assume that other people's actions are all good, and, citing a Gemara in Shabbos (דף קכ"ז:), writes that one who does this will himself be judged favorably by Hashem.
The Beraisa in Maseches Kallah Rabbasi (פרק ט'), however, states that one should always consider another person to be like a thief (at least potentially), which, of course, implies the exact opposite. The Gemara there immediately questions this statement based on another Mishnah in Pirkei Avos (פרק ב' משנה ד') which teaches that one should not judge someone else (negatively, as the Bartenura there points out) until one has been in that situation, implying, again, that one should not suspect another person without knowing all the facts. The Gemara responds that in Pirkei Avos, the Mishnah is talking about a person whom one knows - he should not be judged unfavorably unless all the facts are clear. In Maseches Kallah Rabbasi, however, the Beraisa is referring to a person whom one does not know - he may justifiably be suspected of being wicked.
Rabbeinu Yonah, explaining the Mishnah in Pirkei Avos about judging others favorably (פרק א משנה ו'), writes that one should judge the average person favorably whether one knows him or not, adding in his Shaarei Teshuvah (שער ג' אות רי"ח) that this is required by the Torah, but someone who is known to be a wicked person should always be viewed in a negative or suspicious light. The Klei Yakar, commenting on the Posuk in the Torah (ויקרא י"ט:ט"ו) quoted by the Gemara in Shevuos (דף ל.) as the source for the idea of judging people favorably, notes as well (בד"ה בצדק) that a wicked person should not be judged favorably because the assumption is that he has remained wicked; one is not considered a חושד בכשרים for suspecting such a person because this person is not considered to be among the כשרים. The Bartenura on that Mishnah also writes that physical punishment is afflicted only upon a חושד בכשרים, but one who is חושד a רשע has done nothing wrong. We see from here that this prohibition to be suspicious of other people is not necessarily all-encompassing; there are possible exceptions.
Because of this prohibition, though, it is also necessary for one to avoid doing things that make other people suspicious of him. Rabbeinu Yehudah HaChassid notes in his Sefer Chassidim (סימן מ"ד) that one who causes suspicions to be raised about himself is responsible for the reactions of the people who see him, and hence, their punishment, when applicable. There may, however, be a distinction between an individual and a large group of people because one won't usually suspect an entire group of being sinners. The Gemara in Avodah Zarah (דף מ"ג:) indeed says that the prohibition of being חושד does not apply regarding a group; we need not worry that someone will be חושד an entire group. The Ramo (יורה דעה סימן קמ"א סעיף ד) rules accordingly and thus something forbidden to an individual because it may raise suspicions about him may be permissible for a group.
This point is debated by the Poskim, but the Magen Avraham (אורח חיים סימן רמ"ד ס"ק ח) concurs, explaining that a non-Jew may thus do certain work for a community on Shabbos which he wouldn't be able to do for an individual because there will be no suspicion of an entire community. He therefore rules that strictly speaking, although it has been forbidden for other reasons, a non-Jew may, under certain circumstances, work on building a Shul on Shabbos because nobody will think that the community sinned by hiring him. The Chasam Sofer (שו"ת חתם סופר חלק או"ח סימן ס') suggests that this is true only for something like a Shul where the community participates in it together, but if many people happen to be doing the same thing, each on his own for himself, then there is a problem because they are then like individuals who must avoid suspicious activities, even though there are many of them. The Pardes Yosef on the Posuk in our Parsha quotes that perhaps this is why Moshe was punished despite being suspicious of a group; he was really being suspicious of each of Bnai Yisrael as individuals.