Are the 'esk' sound in grotesque and burlesque pronounced the same?

This answer mostly agrees with the previous ones, but hopefully adding some more context.

My initial thought was, as in the linked podcast, that I never heard of "dropping the k" but after listening to that sound clip I got a better understanding.

Proper English

As has been agreed, and your research showed, the -esque is formally pronounced -esk in both cases. Since the words are of foreign (French) origin, they could inspire a foreign pronunciation, but also in French the -esk is there. (One may think that French always drops the end of the word, and indeed the final 'e' is there to be dropped instead of the 'k'.) Being less used words and spelled a bit odd, there may of course also be reader mistakes or a chance that a person just learned one of the words wrong.

Wordly matters

Languages evolve and linguistics have come up with all sorts of "patterns" to explain how. Phonetics takes interest in the actual pronunciation. I'll mix a bit of those references with intuition to give a possible explaining to your question.

In general, language tends to strive for efficiency. "Food" or "eat" are short words while "culinary experience" is a more cumbersome phrase that has a less basal meaning. The same principle goes for linguistic evolution of a language and phonetic simplification individually applied.

Simplifications

There is a process called Lentition that applies both to languages over time and to individual pronunciation. The basis is that e.g. 'k' is softened to 'g', from there perhaps to a slight 'j' or 'e' or 'r' and eventually dropped entirely. Omission is linguistically named Elision, occurring also for other reasons. I described that as a process over time, but it is probably more so that individual pronunciation takes as much simplification as is "allowed" and the language follows when enough accepts it.

Vowel placement

Construct this experiment: Assume that "to think" in past tense was "thinked". That would be "easily" pronounced with -kt. Now try "thou-kt" and you will probably agree that "thou-gt" has a better flow to it. The more open vowel automatically places the following consonant further back and it becomes softer (as for 'k->g'). Moving the 'g' even further back will softening it into 'gh' (aspiration). Maybe you feel that it's almost gone? Then you have "thought". It's not an accurate description of how the actual word evolved, but showing how phonetic context may affect consonants.

Now try "thinks" and "things", that needs to be clearly different, being different words. Say "He things that..." and I doubt anybody would react, it obviously follows the pattern of softening that we (mostly unknowingly) are used to and accepts as fair use. This is a particularly clever example, because even though you only softened the 'k->g' the 'ng' becomes another sound in context and that sound is already present in /ˈthiŋk/. If analysing a recording, non-linguists would probably argue between "dropped k" or "slight k still there" depending on how you do the transition to 's'. (Also try "I have so many thinks on my mind" and you may generate more surprise, not only because it sounds like a failed aphorism, but because Fortification is not a common simplification.)

Consonant placement

As mentioned, the context is not only vowels, but also consonants. In the podcast they mention having more trouble with "asked" than "asks.". 's' has a high air flow (aspiration) while 'k' and 't' are normally stopping the flow (using the back of the tongue). A double stop -kt is naturally more difficult, but since 'x' is quite frequent in English, the -ks is also a more familiar sound. Standing by the principle of "simplifying the commonly used" indicates that you probably don't say '/k/+/s/' but softening the 'x' with perspiration into 'khs' or 'ghs'.

asks and asked

When this follows another 's', it's clearly less efficient to move the tongue a lot, so you may "emulate" the 'k' "tongue stop of flow" by stopping it in the throat or diaphragmatically instead. I'm not familiar with this being used commonly, instead I would assume there to be a very soft 'gh' lazily in there. I could envision it to happen, and as Mitch says, turn into skipping the stop and towards "an extended /s/". Perhaps pushing more, instead of stopping, to stress two separate s.

In a similar manner, one could try to emulate "k+t" with "t+t". It would probably be accepted, but a "gh+dh" is possibly more efficient (as for "talked"?). With -st being another familiar sound, that seems to be a good choice and (as such) accepted already.

burlesque

I wasn't able to find the video of the actual example. There is no intrinsic reason to drop the ending 'k' from "burlesque", but softening to 'g' is probably not uncommon. If there is something following that makes the 'k' inefficient, as in "burlesques" or "burlesque statue", it may be reduced as mentioned above.

Worldly matters

The phonetic background shows that it is a relaxation from "proper" speech. It would not be the norm in town hall meetings, classic theatres or older radio broadcast, where it was important to speak clearly. When times changed and media is open to anyone (or should I say that clarity and message is less valued than air time itself?), more voices will be heard and the setting is usually more casual and relaxed. Over time this helps "simplification" win ground.

Faster speech is probably more prone to simplification. I don't know any dialect where it is norm. In some sociolects it may be norm to appear relaxed or "lazy" and simplify, in others the simplification may be to appear rushed. Musical style will affect lyrics.

Recordings

When studying audio recordings, it should be noted that the processed speech may have sounded different "live". Filters and noise cancelling can reduce consonants and promote the tonal components, or the details are masked by noise. You may also loose the visual keys that aid hearing.

After listening to the video, I've come to the conclusion that Ross indeed does NOT drop the k noise, it is merely pronounced rapidly.

All words ending with esque are pronounced esk; esque is a suffix derived from Italian meaning resembling/lie something.

To my knowledge, no mainstream dialect drops the "k".

More on http://www.dictionary.com/browse/-esque