OP answered in two parts. Firstly the historical practices of the day, secondly the direct exegesis. The correct translation provided be @PerryWebb is clearly the answer and key to this passage and must taken as a direct reference to the Feast of the Unleavened Bread.
On the translation we have which is 'yeast' and possibly not the best translation, then this might work as a translation within the broader understanding of yeast culture fermentation ...
Historical practices of fermentation Israel versus Egypt To answer @Jess, the confusion regarding the role of 'yeast' (not leavened bread) this would be relevant to Egypt at this point in history. For example, the Book of the Dead for Ka (tomb builder) and Merit (Ka's wife) makes it explicitly clear beer consumption was a dietary staple for the regular working man (women dunno) in Egypt because Ka repeatedly requested this for his afterlife and would have commissioned their Book because he outlived his wife. Ka would have received part of his wages in barley, thus they would have needed yeast to ferment it. I do understand Egyptology is probably out of bounds here, so I'll not expound the example further.
There are few references to beer drinking in the Old Testament, albeit Proverbs 31:4-7 is certainly one of them (implying beer was pretty common) and I'm not aware of any in the New Testament. There are frequent references to wine/vineyards in the Old and New Testament most notably the Last Supper, but key issue is that the fermentation process is very different to beer.
Beer needs a bread style yeast culture, otherwise (generally) its just barley water. Wine works because grape skins carry their own natural yeast (that dusty stuff on grapes is yeast). Thus all thats needed to make wine is to squash 'em and leave it, the yeast on the skin mixes with the sugars within the grapes. I accept a lot of modern wine making (not all) for exact branding and quality control is highly unlikely to rely on the variation from wild yeast.
The relevance is as follows: given the translation we have there would have been little ambiguity (probably none) with the use of a yeast culture as a method of fermentation for Israelites at this point in history. Thus, I take the point that the same yeast used for bread can be used for fermentation - and ancient Egyptians would have likely followed this practice to make beer, but not the Israelites because thats not how they made wine.
The translation 'Beware the leavened bread' (heavily paraphrased) has very different meaning however ...
Biblical exegesis
Just to further @Chengarda comment, the Feast of the Unleavened Bread must be central to the exegesis of the passages of Matt. 16:6, 11; Mark 8:15; Luke 12:1
The presence of bread associated with yeast in any house for anyone (Jew or Gentile) during the Festival resulted in being 'cut-off' from Israel (Exodus 12:15; 19 RSV [below]). Whether the penalty was observed in first century Israel under Roman occupation, who knows, but there must have been significant diligence.
Yeast-risen bread, whilst not prohibited under the Mosaic law, is very bad news indeed if its in the wrong place at the wrong time. Calling someone 'yeast-risen bread', i.e. their teaching, is extreme. If this stuffs in your house at Festival time you're 'cut-off' from Israel.
The precise prohibition on eating yeast-risen bread appears complex and my reading is 3 weeks based Exodus 12 (7 days + 14 days), but could be 14 days in total. The first 7 days of the Feast strictly prohibiting any leavened bread being within the house: I assume versus Exodus 12:19 (RSV) re-refers to the first seven days of this feast and this is where the maximum penalty appears to kick-in.
RSV
14 “This day shall be for you a memorial day, and you shall keep it as
a feast to the Lord; throughout your generations you shall observe it
as an ordinance for ever. 15 Seven days you shall eat unleavened
bread; on the first day you shall put away leaven out of your houses,
for if any one eats what is leavened, from the first day until the
seventh day, that person shall be cut off from Israel.
17 And you shall observe the
feast of unleavened bread, for on this very day I brought your hosts
out of the land of Egypt: therefore you shall observe this day,
throughout your generations, as an ordinance for ever. 18 In the first
month, on the fourteenth day of the month at evening, you shall eat
unleavened bread, and so until the twenty-first day of the month at
evening. 19 For seven days no leaven shall be found in your houses;
for if any one eats what is leavened, that person shall be cut off
from the congregation of Israel, whether he is a sojourner or a native
of the land. 20 You shall eat nothing leavened; in all your dwellings
you shall eat unleavened bread.”
Summary
In my personal opinion, this is a poor translation from two perspectives:
The translation is antiquated, given the historic application of yeast culture fermentation is not generally understood (why should it be). Whilst the translators from William Tyndale onwards might have had a strong understanding of this, this cannot be assumed.
Furthermore the translation takes us away from the shear intensity of Jesus' words. This is gut-punch to the Pharisees. There is no way anyone should go near their teaching, which was central to their very their role.
I've done the tour BTW to pre-empt the welcome message.