An excellent puzzle from Mr E, even if slightly unseasonal although publishing it at the more obvious time of year might have made the theme too easy to guess.
The first thing to notice is that all across answers are one letter longer than their grid entries, so must be modified somehow before entry. By a process of elimination, it turns out that one letter must be removed from each answer and, taken in turn, these spell out AULD ACQUAINTANCE, part of a quotation from Auld Lang Syne by Robert Burns - "Should auld acquaintance be forgot and never brought to mind". The second part of the question is answered in the affirmative by adding NEVER to the answer for 21D producing the entry NEVER MIND.
In the following explanations, the letters to be removed from across answers before entry are shown highlighted in bold and the down answers are followed by the number of the clue where its definition is to be found.
ACROSS
1 Tragedian ignoring one is awkward in part (14, two words)
STAGE DIRECTION (anag of TRAGEDIAN minus AN (one) in SECTION (part))
10 Unheated building - a place for putting game (10)
GREENHOUSE (GREEN (place for putting) + HOUSE (game - bingo))
12 Language found in eastern land (7)
BENGALI (ENG (language - English) in BALI (eastern land) &lit)
13 Film producer and director is not unknown (6)
DISNEY (D (director) + IS + NE (not) + Y (unknown))
14 A cake, and in due course a second demolished (9, two words)
SODA SCONE (SO (in due course) + anag of A SECOND)
17 Being so, you would see a rook in the meadow earlier (9)
CLEAR-EYED (C (see) + A + R (rook) in LEA (meadow) + YED (you would - ye'd)
semi &lit)
18 Answering question in re an armband (6)
TORQUE (TO (answering) + QU (question) in RE)
19 Provider of birds (some French) over a period of days (8)
TRUELOVE (cryptic ref to carol Twelve Days of Christmas) Unusual to see this
kind of Times-style cryptic def used in a Listener
20 Most of important employees taken in by profitable swindle (8)
FAKEMENT (KEY (important) minus last letter + MEN (employees) in FAT
(profitable))
22 Variety of fruit ripens late, developing with no spoilt part (6)
NELIES (comp anag RIPENS LATE minus PART semi &lit)
25 We all remember this hard earth, in version of Spooner (9)
NOOSPHERE (anag of SPOONER round H (hard) + E (earth))
29 Here's to Ximenes' cryptic book (9)
SIXTEENMO (anag of TO XIMENES)
30 The wrong way to capture an animal for fur (6)
MARTEN (NET (capture) + RAM (animal) all rev)
31 Incurring the chance of the king being taken by rebellion (7)
RISKING (RISING (rebellion) round K (king))
32 Venomous one makes dancer feel bad (10)
FER-DE-LANCE (anag of DANCER FEEL)
33 Ordinary or choice (of pieces of ground) (14)
COMMON-OR-GARDEN (choice of pieces of ground)
DOWN
1 Buses on time? Doesn't happen! (6)
SUBSET [26] (anag of BUSES + T (time))
2 The old punishment is death by drowning (9)
THEOTOKOS [17] (THE + O (old) + TOKOS (punishment))
3 Apple garden in disarray (6)
GANDER [27] (anag of GARDEN)
4 Open Ohio map from the bottom (5)
DRACO [24] (O (Ohio) + CARD (map) all rev)
5 Stop a bill that's brought up (6)
RENNET [3] (TENNER (bill) rev)
6 In Caen hear several repetitions of CD? (8)
ENHEARSE [8] (hidden)
7 Well, not according to the officials on the field (4)
CHIP [28] (CHIPPER (well) minus PER (according to))
8 Playwright's to enclose directions in Italian (4)
ISN'T [1] (NS (directions) in IT (Italian))
9 Ban the bomb? One day, possibly (6)
NOYADE [2] (anag of ONE DAY)
11 Succeeded in tossing eels to large fish? (7)
OSSELET [20] (anag of EELS TO round S (succeeded))
15 Pack animal following motorway in Germany (4)
MIND [21] (M (motorway) + IN + D (Germany))
16 Around small port is an ancient region of scavengers (7)
SUMPTER [15] (SUMER (ancient region) round PT (port))
17 Mother of God, is the speaker in undies? (8)
LINESMEN [7] (LINEN (undies) round S (is) + ME (the speaker))
20 Plastic laminate, not a hard substance (6)
FORMIC [16] (FORMICA (plastic laminate) minus A)
21 The artist has lost his head - look out (6)
TRILLO [6] (UTRILLO (artist) minus first letter)
23 Incompetent person not caught by anyone (6)
LUNKER [11] (CLUNKER (incompetent person) minus C (caught))
24 Cousin to monitor record of progress in hospital (6)
SLOGAN [9] (SAN (hospital) round LOG (record of progress))
26 Good government: some elements are not part of it (5)
BEING [23] (BEIN (good) + G (government))
27 Has the Earl married? The fool! (4)
STEM [5] (S (has) + T (the) + E (Earl) + M (married))
28 A shot from close to green in middle of round one (4)
UNDO [4] (hidden)