Short answer: There are probably no languages "which because of their structure do not allow for the use of sarcasm or irony" since much sarcasm doesn't depend on certain features/structures. But it is interesting to look at the relation between linguistic features and sarcasm.
Longer musings: There are certainly kinds of sarcasm which are facilitated by the features of a language. Languages like Japanese and Korean have highly codified systems of politeness/formality. The verb forms (among other things) we use change according to our interlocutor (friend vs. boss), the situation (chatting with co-workers over lunch vs. presenting to same co-workers in a meeting), etc.
Imagine a situation where my friend is annoyingly chattering away about something uninteresting to me. I could exasperatedly say:
Aa sou desu ka
Ah like.that COP.POL Q
‘Oh, is that so?’
using a polite sentencing ending that I wouldn’t normally use to my friend. On the surface I show interest in what my friend is saying; my disinterest is sarcastically implied.
Wetzel (2004: 97) gives the following example (again, in Japanese) of a wife responding to her husband who has asked her where to find something (my gloss):
Genkan desu yo, itsumo-to onaji
Entranceway COP.POL EMPH always-with same
‘It’s in the entranceway as usual.’
She analyses the example as follows:
Such uses of polite language would rarely be interpreted to indicate
that this wife views her husband as her lord or master. Far from it;
most Japanese would take [this response] as symptomatic of
anger, as polite style between husband and wife is indicative of
sarcasm or of psychological/emotional distance (pp. 97-8).
An example from Korean would be:
nwee nwee alkesssupnita
‘Yes, yes, understood.’
where nwee is a sarcastic elongation of ne 'yes' with characteristic pronunciation. The forms for 'yes' and 'understand' are very polite. On the surface this expression seems to convey the speaker's polite acquiescence, but it is conventionally understood to be a sarcastic ignoring/dismissal (cf. lang-8). The polite forms here are fixed -- you would use this expression as is to your friend with whom you normally speak casually.
It’s interesting to consider what features of English (or your own native language) might stand out like this. Perhaps the Split Interrogative in English -- a WH-question followed immediately by a sarcastic answer -- would seem like a particularly English kind of sarcasm to speakers of languages without a similar construction:
You don’t trust a man that won’t drink with you? That doesn’t even
make sense. What is this, the Wild West? (Michaelis & Feng 2015: 153)
So features of a language, which are not found in all languages, can be used to sarcastic effect. But as Cerberus says in comments, most sarcasm is not conventional; we use the speaker’s tone of voice, as well as our knowledge of the context, of the speaker, and so on, to discern that they are implying something other than the literal meaning of their utterance. (You can bring Grice’s maxims, indirect speech acts, etc. in here if you want.) A few examples:
Just so strong. What a strong player. I really have respect for him
and how he's played this game. (Dan ‘Artosis’ Stemkoski
[https://www.twitch.tv/videos/558906301] after losing to a strategy he
Juna oficisto denove petis liberan tempon. La ĉefo diris al li: Mi
vere estas scivola, kion vi nun eltrovis. Se mi ne eraras, vi jam
kvarfoje enterigis vian avinon en ĉu tiu jaro. Ĉu ŝi denove mortis?
(Miyamoto 1962: 8)
‘A young office worker applied for time off again. The
boss said to him, “I’m really curious what you’ve found now. If
I’m not mistaken, you’ve already buried your grandmother four times
this year. Did she die again?”’
Japanese (start of a sarcastic response after being called an 'idiot'):
Baka da kara tensai no omae ga oshiete hoshii
idiot COP so genius GEN you(rude) NOM teach want
'I'm an idiot so I want you, the genius, to explain.'
Some cultures use this kind of sarcasm more than others. My personal experience, and plenty of personal accounts on the internet, tell me that sarcasm is less common in Japan than in the UK or US, for example. But since it is possible to convey sarcasm not by any particular feature of a language but simply through the implicature created by the difference between the literal meaning of an utterance and our expectations or understanding of the situation, it is hard to imagine a language in which sarcasm is impossible due to the presence or lack of "certain features or structures".
Features of a language (which other languages may not have) such as a highly codified system of politeness in Japanese and Korean are certainly exploited in sarcasm.
Sometimes constructions have a conventionally understood sarcastic meaning, as in the English Split Interrogative or the Korean sarcastic response example.
But most sarcasm is not conventional in this way. With our contextual knowledge, etc., we read sarcastic meanings into otherwise ordinary utterances.
And so, while some cultures may use this kind of sarcasm less than others, it is difficult to imagine a language in which the presence or lack of certain structures makes sarcasm altogether impossible.
Michaelis, L. A. & Feng, H. (2015). What is this, sarcastic syntax?
Constructions and Frames, 7(2), 148-180.
Miyamoto, M. (trans./ed.) (1962). Facilaj legaĵoj. Tokyo:
Wetzel, P. J. (2004). Keigo in modern Japan: Polite language from
Meiji to the present. Honolulu: University of Hawai’i Press.