You have to declare a font encoding that supports cyrillic, for instance T2A. Then, as the commands for producing the characters don't have (for efficiency reason) a default encoding, you have to add it for the characters you need.
\documentclass{article}
\usepackage[T2A,T1]{fontenc}
\usepackage[utf8]{inputenc}
\DeclareTextSymbolDefault{\cyre}{T2A}
\DeclareTextSymbolDefault{\cyrf}{T2A}
\begin{document}
Adding the cyrillic character е in this sentence doesn't crash
the document so I don't have to delete the character е.
It doesn't crash even with ф.
\end{document}
The names of the letters are easy, usually. For the uppercase Е it's \CYRE and so on. You find them in the file t2aenc.def, lines 102–165.

A hack for enabling all letters in one swoop:
\documentclass{article}
\usepackage[T2A,T1]{fontenc}
\usepackage[utf8]{inputenc}
\begingroup
\makeatletter
\providecommand\@gobblethree[3]{}
\let\DeclareFontEncoding\@gobbletwo
\let\DeclareFontSubstitution\@gobblefour
\let\DeclareTextAccent\@gobblethree
\let\@tempa\@empty
\renewcommand\DeclareTextCommand[2]{\renewcommand\@tempa}
\let\DeclareTextComposite\@gobblefour
\def\DeclareTextSymbol#1{\expandafter\decidecyr\string#1....\decidecyr{#1}}
\def\decidecyr#1#2#3#4#5\decidecyr#6{%
\lowercase{\def\@tempa{#1#2#3#4}}\edef\@tempb{\string\cyr}%
\ifx\@tempa\@tempb
\expandafter\declarecyr\expandafter#6%
\else
\expandafter\@gobbletwo
\fi
}
\newcommand\declarecyr[3]{%
\toks@=\expandafter{\the\toks@\DeclareTextSymbolDefault{#1}{T2A}}}
\toks@={}
\input{t2aenc.def}
\expandafter\endgroup\the\toks@
\begin{document}
Adding the cyrillic character е in this sentence doesn't crash
the document so I don't have to delete the character е.
It doesn't crash even with ф.
\end{document}