Related to Are there any jokes in tanach?, what puns exist in Tanach?
The Ibn Ezra, in his commentary to Bereishis 2:25, gives the following examples:
כה וַיִּהְיוּ שְׁנֵיהֶם עֲרוּמִּים, הָאָדָם וְאִשְׁתּוֹ; וְלֹא, יִתְבֹּשָׁשׁוּ.
א וְהַנָּחָשׁ, הָיָה עָרוּם, מִכֹּל חַיַּת הַשָּׂדֶה, אֲשֶׁר עָשָׂה יְהוָה אֱלֹהִים; וַיֹּאמֶר, אֶל-הָאִשָּׁה, אַף כִּי-אָמַר אֱלֹהִים, לֹא תֹאכְלוּ מִכֹּל עֵץ הַגָּן.
One means "naked," and one means "cunning."
טז וַיֹּאמֶר שִׁמְשׁוֹן--בִּלְחִי הַחֲמוֹר, חֲמוֹר חֲמֹרָתָיִם; בִּלְחִי הַחֲמוֹר, הִכֵּיתִי אֶלֶף אִישׁ.
One means "donkey," and one means "many heaps."
ד וַיְהִי-לוֹ שְׁלֹשִׁים בָּנִים, רֹכְבִים עַל-שְׁלֹשִׁים עֲיָרִים, וּשְׁלֹשִׁים עֲיָרִים, לָהֶם; לָהֶם יִקְרְאוּ חַוֹּת יָאִיר, עַד הַיּוֹם הַזֶּה, אֲשֶׁר, בְּאֶרֶץ הַגִּלְעָד.
One means "donkeys" and one means "cities."
He also asserts in his commentary to Shemos 22:5 that this is another example of a pun:
ד כִּי יַבְעֶר־אִישׁ שָׂדֶה אוֹ־כֶרֶם וְשִׁלַּח אֶת־בְּעִירֹה וּבִעֵר בִּשְׂדֵה אַחֵר מֵיטַב שָׂדֵהוּ וּמֵיטַב כַּרְמוֹ יְשַׁלֵּֽם׃
ה כִּֽי־תֵצֵא אֵשׁ וּמָֽצְאָה קֹצִים וְנֶֽאֱכַל גָּדִישׁ אוֹ הַקָּמָה אוֹ הַשָּׂדֶה שַׁלֵּם יְשַׁלֵּם הַמַּבְעִר אֶת־הַבְּעֵרָֽה׃
In the first verse, they all are related to the word "animal," while in the second it refers to fire.
This article, by Professor Gary Rendsburg of Cornell University, provides a number of examples of puns (some of them bilingual) from throughout Tanakh. As you can see from the final page, it is an article within a collection that deals explicitly with this issue.
My personal favourite from the ones that he cites, and he cites many, is a Hebrew/Greek bilingual pun in Proverbs 31:27. Rather than say צופה הליכות ביתה ("she watches over the ways of her household"), a more unusual form of the verb, צופיה, is used, and one which allows the author to pun on the Greek word for wisdom (sofia).
Most importantly, the article is replete with references to other texts (many of them by the same author, though not all of them), which explore other more specific examples of biblical Hebrew wordplay.
Here are two of my favorite:
Bamidbar 24:10–11: והנה ברכת ברך זה שלש פעמים׃ ועתה ברח־לך אל־מקומך; you have continually blessed (barekh) them three times. Now flee (b’raḥ) to your place.
Devarim 11:16–17: ועבדתם אלהים אחרים … ואבדתם מהרה; lest you serve (va‘avadtem) strange gods … you will quickly be banished (va’avadtem).
Moshe rabbeinu made a couple, according to Rashi:
Shmos 32:18 וַיֹּאמֶר, אֵין קוֹל עֲנוֹת גְּבוּרָה, וְאֵין קוֹל, עֲנוֹת חֲלוּשָׁה; קוֹל עַנּוֹת, אָנֹכִי שֹׁמֵעַ.
(The dagesh on עַנּוֹת drives the point home.)
Bamidbar 21:9 וַיַּעַשׂ מֹשֶׁה נְחַשׁ נְחֹשֶׁת Which, according to Rashi, was a bit of a pun.
Yeshaya 5:7 וַיְקַו לְמִשְׁפָּט וְהִנֵּה מִשְׂפָּח, לִצְדָקָה וְהִנֵּה צְעָקָה. Rashi explains מִשְׁפָּט/מִשְׂפָּח and צְדָקָה/צְעָקָה are used because they sound similar.
I've always been enamored with Bamidbar 19:17: וְנָתַן עָלָיו מַיִם חַיִּים, אֶל-כֶּלִי. I'm waiting to be at a meal with a person name Kelly, who asks someone (not next to her) to pass the Mayim Chaim (brand of soda pop). When the person passes it over the man sitting next to Kelly, he will place it on his head, reciting this verse.
I also like Bamidbar 22:30 הֲלוֹא אָנֹכִי אֲתֹנְךָ אֲשֶׁר-רָכַבְתָּ עָלַי מֵעוֹדְךָ עַד-הַיּוֹם הַזֶּה
I hear it as "Hello?! Am i not your donkey?! ..." This works for any הֲלוֹא, but i like that one best.
What of Bereishit 29:10 and 11?
וַיַּשְׁקְ, אֶת-צֹאן לָבָן אֲחִי אִמּוֹ. וַיִּשַּׁק יַעֲקֹב, לְרָחֵל And he watered the sheep of Lavan, his uncle. And Jacob kissed Rachel
The verbs here are very similar (only the vowels change). This makes it both a pun and a trap for bad baalei kriah.
Most of these are from Chumash. My favorite is in Shmuel Aleph 15:14. Shaul was commanded to wipe out Amalek, including all of their animals. Shmuel came to see how Shaul had done. Shaul greeted Shmuel delighted, implying he had completed his command. Shmuel responds:
וַיֹּ֣אמֶר שְׁמוּאֵ֔ל וּמֶ֛ה קֽוֹל־הַצֹּ֥אן הַזֶּ֖ה בְּאָזְנָ֑י וְק֣וֹל הַבָּקָ֔ר אֲשֶׁ֥ר אָנֹכִ֖י שֹׁמֵֽעַ׃
“Then what,” demanded Samuel, “is this bleating of sheep in my ears, and the lowing of oxen that I hear?”
The word ומה has the tevir trop symbol. When the word is read with the trop, it sounds like the noise a sheep makes.
Edit: I just found this was posted as an answer here. I feel it's more of a pun than a joke.
However, I found another in Chumash no one seems to have mentioned.
וַאֲנִ֞י נָתַ֧תִּֽי לְךָ֛ שְׁכֶ֥ם אַחַ֖ד עַל־אַחֶ֑יךָ אֲשֶׁ֤ר לָקַ֙חְתִּי֙ מִיַּ֣ד הָֽאֱמֹרִ֔י בְּחַרְבִּ֖י וּבְקַשְׁתִּֽי׃ (פ)
And now, I assign to you one portion more than to your brothers, which I wrested from the Amorites with my sword and bow.”
Yaakov tells Yosef he is giving him a "Shechem" over his brothers. Onkelos and Rashi explain this to mean portion, since the number that follows it is masculine, and cities are feminine. However, what is the portion he is getting? Rashi says the city of Shechem!
I'm wondering if אברהם gave 7 sheep to אבימלך as witness that the wells belong to אברהם (Breishis 21:30), and then אברהם and אבימלך swore their treaty, and באר שבע is named because of the swearing (שבועה). I wonder if אברהם picked 7 animals because the word שבע with a סגול sounds like שבע with a פתח, and it's just a play on words.
Also ובן משק ביתי הוא דמשק אליעזר (Breishis 15:2). Depending on how you translate the words בן משק and דמשק, if they aren't related on a pshat level, but are put together because they sound similar, that may be a play on words.
Edit: I thought of another possibility. יצחק is named so because of the laughing of אברהם and שרה, which are different but share the meaning "laughing". Then שרה says "שחוק עשה לי", a different type of laughter. Then ישעמאל is מצחק - something that is also a very different type of laughter.
Ex 13:14 ויהי בשלח פרעה את העם ולא נחם אלהים דרך ארץ פלשתים כי קרוב הוא כי אמר אלהים פן ינחם העם בראתם מלחמה ושבו מצרימה
What about Shemot 14:11, where they ask Moshe Rabbenu: “Weren’t there any burial-places in Egypt, so you took us to die here?”
Are not the names of most of Jacob's sons technically puns?