It is halachically forbidden for the gabbai to honor an individual with Hagbah if he knows he will not be able to do it properly. It is imperative that the congregation be able to see the writing in the Sefer Torah well, and therefore the Torah must be opened enough and for long enough to see the writing in the Torah. If the person being called up is unaware of how to do Hagbah properly, he should be shown how to do so prior to being called up. If he is not confident he can do it properly, someone who is prepared to do Hagbah properly should receive the Hagbah. If it is difficult for a person, or he is physically incapable of doing Hagbah properly, it is forbidden to give him the Aliyah.
This teaching is based on the Sha'arei Ephraim (Rav Ephraim Zalman Margolis) and is quoted in the "Mishna B'rura" (Siman 147, S"K 7).
The below is a translation from the "Sha'arei Ephraim" on this matter:
The gabbai is not allowed to honor with the mitzva of hagbah someone who has very shaky hands or an elderly person or a weak person, for whom it is known that he will pick it up in such a manner so that he can sit down immediately, for he cannot lift the Torah with strength for a period of time facing the congregation so that they could take a good look as a result of him turning around slowly. Also, the person himself should avoid doing Hagbah, since he is unable to fulfill the mitzva as it was instituted, and Chazal were very strict about this that the mitzvos be done by someone who is able to fulfill it as it was instituted.
Sha'arei Ephraim 10:14:
ואין הסגן רשאי לכבד עם מצות הגבהה למי שידיו רותתין הרבה או לזקן ותש כח שידוע שהוא מגביהו לשבת מיד כי לא בכח יגבר להחזיקה זמן מה כלפי העם שיוכלו להסתכל היטב ע"י שהוא פונה לסובב בנחת וגם האיש ההוא ימנע נפשו מזה אחרי שא"א לו לעשות המצוה כתיקונה וחז"ל החמירו בזה מוטב שתעשה המצוה ע"י מי שיש סיפוק בידו לעשותה כתיקנה
Mishna Berura 147:7 follows this ruling:
שאין הסגן רשאי לכבד במצות הגבהה למי שידיו רותתין הרבה או לזקן ותש כח שידוע שהוא מגביהו לשבת מיד כי אין לו כח להחזיקה זמן מה כלפי העם שיוכלו להסתכל היטב וגם האיש ההוא צריך למנוע נפשו מזה