The NET Bible has a long note dealing with this verse. They decided to retain the reading.
Mark 7:3 (For the Pharisees and all the Jews do not eat unless they perform a ritual washing, holding fast to the tradition of the elders. 7:4 And when they come from the marketplace, they do not eat unless they wash. They hold fast to many other traditions: the washing of cups, pots, kettles, and dining couches.)
Their reasoning:
tc Several important witnesses (Ì45vid א B L Δ 28* pc) lack “and dining couches” (καὶ κλινῶν, kai klinwn), while the majority of mss (A D W Θ Ë1,13 33 Ï latt) have the reading. Although normally the shorter reading is to be preferred, especially when it is backed by excellent witnesses as in this case, there are some good reasons to consider καὶ κλινῶν as authentic: (1) Although the addition of κλινῶν could be seen as motivated by a general assimilation to the purity regulations in Lev 15 (as some have argued), there are three problems with such a supposition: (a) the word κλίνη (klinh) does not occur in the LXX of Lev 15; (b) nowhere in Lev 15 is the furniture washed or sprinkled; and (c) the context of Lev 15 is about sexual impurity, while the most recent evidence suggests that κλίνη in Mark 7:4, in keeping with the other terms used here, refers to a dining couch (cf. BDAG 549 s.v. κλίνη 2). Thus, it is difficult to see καὶ κλινῶν as a motivated reading. (2) κλίνη, though a relatively rare term in the NT, is in keeping with Markan usage (cf. Mark 4:21; 7:30). (3) The phrase could have been dropped accidentally, at least in some cases, via homoioteleuton. (4) The phrase may have been deliberately expunged by some scribes who thought the imagery of washing a dining couch quite odd. The longer reading, in this case, can thus be argued as the harder reading. On balance, even though a decision is difficult (especially because of the weighty external evidence for the shorter reading), it is preferable to retain καὶ κλινῶν in the text.
To put their argument in layman's terms, even though some important manuscripts omit the clause, many other important ones have them. One of the rules of textual criticism is, all other things being equal, to take the harder reading as the original. That is, a copyist is more likely to make things easier to understand than to make it harder. One exception is when the scribe is purposefully corrupting a manuscript (An answer to your question on the doubled amen in Romans 16 contains a good example of Marcion or at least Marcion's followers corrupting the text).
One question text critics would ask on this verse is "would a scribe be more likely to remove καὶ κλινῶν or add καὶ κλινῶν?" Removal could come about in several ways. The first part of the Pharisee's tradition comes from Leviticus 15, but Leviticus 15 does not mention furniture. Because of that, a scribe could easily think that καὶ κλινῶν was added and remove it (thinking that he is restoring the text). It could have been accidental because there is a near rhyme in the verse. After writing καὶ χαλκίων ("and kettles") the scribe accidentally skipped καὶ κλινῶν ("and dining couches").
They also note that Mark uses κλίνη (a form of κλινῶν) twice. Also, since Leviticus 15 isn't talking about furniture, it is hard to see a scribe wanting to add καὶ κλινῶν (adding it is not a motivated reading and is thus less likely to happen).
Edited: As asked in the OP, many modern translations omit the two words. As this seems to go against the rules of textual criticism, why would they do that?
Answer: In the end, it comes down to a judgment call by different committees of human beings. One set of them is wrong, obviously, but that does not mean that particular set "broke the rules." They weighed things differently and came to different conclusions.
The translators of those versions weighed the evidence differently and concluded that the original more likely omitted the two words. For example, the NET translators said the external evidence for omitting it is "weighty" even though they decided to retain it (and their final decision was "difficult"). Another committee could easily conclude that those early manuscripts (including both Sinaiticus and Vaticanus) are preferable. They might reason that it must have been added because Leviticus 15 does not mention dining couches.
One thing I would want to look into is the possibility of the tradition of washing the dining couches being added during this time. At the time of Leviticus, dining couches weren't used. In the first century, they were. The Pharisees were known for expanding the traditions to keep up with contemporary times (this is called "Fence Building" and is a very ancient and respected practice). You wash all of these other items used when eating, washing the dining couch is only a small step.
