Why do many modern Bibles omit the words "take up the cross" in Mark 10:21?
Most modern Bibles are translations based on a Greek critical text like the NA27 or UBS 4. These are eclectic texts compiled by scholars by examining all the available manuscripts and applying principles of textual criticism to try and arrive at a reproduction of the original autographs.
In the case of Mark 10:21, these scholars consider the weight of evidence to indicate that the phrase αρας τον σταυρον ("taking up the cross") to be an addition by scribes who have borrowed the phrase from Mark 8:34. This is based on the weight of manuscripts where the phrase is missing. Most important among these are Sinaiticus and Vaticanus which are among the oldest and most complete manuscripts available and which modern scholars consider to be some of the best surviving manuscripts. But also a number of other important manuscripts - C D Δ Θ Ψ 0274 - do not have the phrase in Mark 10:21.
While there are a couple of older manuscripts that do include this phrase (notably Alexandrinus and Washingtonius) in the judgment of the scholars building the critical text, the textual evidence points pretty strongly in favor of the shorter reading:
The Textus Receptus, following A and many minuscules, adds a gloss from 8:34, ἄρας τὸν σταυρόν. The shorter text is strongly supported by א B C D Δ Θ Ψ al.1
The WH NU reading has superior testimony among early and diverse manuscripts. At the end of this verse, several scribes add "take up the cross," borrowed from 8:34.2
1 Metzger, B. M., United Bible Societies. (1994). A textual commentary on the Greek New Testament, second edition a companion volume to the United Bible Societies’ Greek New Testament (4th rev. ed.) (p. 89). London; New York: United Bible Societies.
2 Comfort, P. W. (2008). New Testament Text and Translation Commentary: Commentary on the Variant Readings of the Ancient New Testament Manuscripts and How They Relate to the Major English Translations (p. 138). Carol Stream, IL: Tyndale House Publishers, Inc.
As shown above, the preponderance of Greek evidence favors the omission of "take up your cross". I would like to add the Latin evidence, from my "A reconstruction of the original Latin of Mark's gospel" (on the internet). The other 7 Old Latin manuscripts omit it, a q have it, but in different wording (being different translations from the Greek) and at different places, q clumsily inserted at the end of the verse, showing it is clearly a later addition. The Latin is clearly in favor of its omission.